diff --git a/src/main/java/com/conveyal/analysis/components/broker/Broker.java b/src/main/java/com/conveyal/analysis/components/broker/Broker.java
index 1eeb93641..2aea67dc2 100644
--- a/src/main/java/com/conveyal/analysis/components/broker/Broker.java
+++ b/src/main/java/com/conveyal/analysis/components/broker/Broker.java
@@ -48,36 +48,41 @@
 /**
  * This class distributes the tasks making up regional jobs to workers.
  * <p>
- * It should aim to draw tasks fairly from all organizations, and fairly from all jobs within each
- * organization, while attempting to respect the transport network affinity of each worker, giving
- * the worker tasks that require the same network it has been using recently.
+ * It respects the declared transport network affinity of each worker, giving the worker tasks that
+ * relate to the same network it has been using recently, and is therefore already loaded in memory.
  * <p>
- * Previously workers long-polled for work, holding lots of connections open. Now they short-poll
- * and sleep for a while if there's no work. This is simpler and allows us to work withing much more
- * standard HTTP frameworks.
+ * In our initial design, workers long-polled for work, holding lots of connections open. This was
+ * soon revised to short-poll and sleep for a while when there's no work. This was found to be
+ * simpler, allowing use of standard HTTP frameworks instead of custom networking code.
  * <p>
- * The fact that workers continuously re-poll for work every 10-30 seconds serves as a signal to the
- * broker that they are still alive and waiting. This also allows the broker to maintain a catalog
- * of active workers.
+ * Issue conveyal/r5#596 arose because we'd only poll when a worker was running low on tasks or
+ * idling. A worker could disappear for a long time, leaving the backend to assume it had shut down
+ * or crashed. Workers were revised to poll more frequently even when they were busy and didn't
+ * need any new tasks to work on, providing a signal to the broker that they are still alive and
+ * functioning. This allows the broker to maintain a more accurate catalog of active workers.
  * <p>
- * Because (at least currently) two organizations never share the same graph, we can get by with
- * pulling tasks cyclically or randomly from all the jobs, and actively shape the number of workers
- * with affinity for each graph by forcing some of them to accept tasks on graphs other than the one
- * they have declared affinity for.
+ * Most methods on this class are synchronized because they can be called from many HTTP handler
+ * threads at once (when many workers are polling at once). We should occasionally evaluate whether
+ * synchronizing all methods to make this threadsafe is a performance issue. If so, fine-grained
+ * locking may be advantageous, but as a rule it is much harder to design, test, and maintain.
  * <p>
- * This could be thought of as "affinity homeostasis". We  will constantly keep track of the ideal
+ * Workers were originally intended to migrate from one network to another to handle subsequent jobs
+ * without waiting for more cloud compute instances to start up. In practice we currently assign
+ * each worker a single network, but the balance of workers assigned to each network and the reuse
+ * of workers could in principle be made more sophisticated. The remainder of the comments below
+ * provide context for how this could be refined or improved.
+ *
+ * Because (at least currently) two organizations never share the same graph, we could get by with
+ * pulling tasks cyclically or randomly from all the jobs, and could actively shape the number of
+ * workers with affinity for each graph by forcing some of them to accept tasks on graphs other than
+ * the one they have declared affinity for. If the pool of workers was allowed to grow very large,
+ * we could aim to draw tasks fairly from all organizations, and fairly from all jobs within each
+ * organization.
+ * <p>
+ * We have described this approach as "affinity homeostasis": constantly keep track of the ideal
  * proportion of workers by graph (based on active jobs), and the true proportion of consumers by
  * graph (based on incoming polling), then we can decide when a worker's graph affinity should be
- * ignored and what it should be forced to.
- * <p>
- * It may also be helpful to mark jobs every time they are skipped in the LRU queue. Each time a job
- * is serviced, it is taken out of the queue and put at its end. Jobs that have not been serviced
- * float to the top.
- * <p>
- * Most methods on this class are synchronized, because they can be called from many HTTP handler
- * threads at once.
- *
- * TODO evaluate whether synchronizing all methods to make this threadsafe is a performance issue.
+ * ignored and what graph it should be forced to.
  */
 public class Broker implements Component {
 
@@ -100,7 +105,15 @@ public interface Config {
     private final ListMultimap<WorkerCategory, Job> jobs =
             MultimapBuilder.hashKeys().arrayListValues().build();
 
-    /** The most tasks to deliver to a worker at a time. */
+    /**
+     * The most tasks to deliver to a worker at a time. Workers may request less tasks than this, and the broker should
+     * never send more than the minimum of the two values. 50 tasks gives response bodies of about 65kB. If this value
+     * is too high, all remaining tasks in a job could be distributed to a single worker leaving none for the other
+     * workers, creating a slow-joiner problem especially if the tasks are complicated and slow to complete.
+     *
+     * The value should eventually be tuned. The current value of 16 is just the value used by the previous sporadic
+     * polling system (WorkerStatus.LEGACY_WORKER_MAX_TASKS) which may not be ideal but is known to work.
+     */
     public final int MAX_TASKS_PER_WORKER = 16;
 
     /**
@@ -317,9 +330,13 @@ public void createWorkersInCategory (WorkerCategory category, WorkerTags workerT
 
     /**
      * Attempt to find some tasks that match what a worker is requesting.
-     * Always returns a list, which may be empty if there is nothing to deliver.
+     * Always returns a non-null List, which may be empty if there is nothing to deliver.
+     * Number of tasks in the list is strictly limited to maxTasksRequested.
      */
-    public synchronized List<RegionalTask> getSomeWork (WorkerCategory workerCategory) {
+    public synchronized List<RegionalTask> getSomeWork (WorkerCategory workerCategory, int maxTasksRequested) {
+        if (maxTasksRequested <= 0) {
+            return Collections.EMPTY_LIST;
+        }
         Job job;
         if (config.offline()) {
             // Working in offline mode; get tasks from the first job that has any tasks to deliver.
@@ -335,7 +352,10 @@ public synchronized List<RegionalTask> getSomeWork (WorkerCategory workerCategor
             return Collections.EMPTY_LIST;
         }
         // Return up to N tasks that are waiting to be processed.
-        return job.generateSomeTasksToDeliver(MAX_TASKS_PER_WORKER);
+        if (maxTasksRequested > MAX_TASKS_PER_WORKER) {
+            maxTasksRequested = MAX_TASKS_PER_WORKER;
+        }
+        return job.generateSomeTasksToDeliver(maxTasksRequested);
     }
 
     /**
diff --git a/src/main/java/com/conveyal/analysis/components/broker/Job.java b/src/main/java/com/conveyal/analysis/components/broker/Job.java
index fa2c2ca66..e49cf40a9 100644
--- a/src/main/java/com/conveyal/analysis/components/broker/Job.java
+++ b/src/main/java/com/conveyal/analysis/components/broker/Job.java
@@ -177,7 +177,7 @@ public boolean isErrored () {
     /**
      * @param maxTasks the maximum number of tasks to return.
      * @return some tasks that are not yet marked as completed and have not yet been delivered in
-     *         this delivery pass.
+     *         this delivery pass. May return an empty list, but never null.
      */
     public List<RegionalTask> generateSomeTasksToDeliver (int maxTasks) {
         List<RegionalTask> tasks = new ArrayList<>(maxTasks);
diff --git a/src/main/java/com/conveyal/analysis/components/broker/WorkerCatalog.java b/src/main/java/com/conveyal/analysis/components/broker/WorkerCatalog.java
index c91ca7ae8..18ed6534f 100644
--- a/src/main/java/com/conveyal/analysis/components/broker/WorkerCatalog.java
+++ b/src/main/java/com/conveyal/analysis/components/broker/WorkerCatalog.java
@@ -1,6 +1,7 @@
 package com.conveyal.analysis.components.broker;
 
 import com.conveyal.r5.analyst.WorkerCategory;
+import com.conveyal.r5.analyst.cluster.AnalysisWorker;
 import com.conveyal.r5.analyst.cluster.WorkerStatus;
 import com.google.common.collect.HashMultimap;
 import com.google.common.collect.Multimap;
@@ -21,7 +22,8 @@
  */
 public class WorkerCatalog {
 
-    public static final int WORKER_RECORD_DURATION_MSEC = 2 * 60 * 1000;
+    /** If a worker says it will poll every s seconds or less, wait s plus this number before considering it gone. */
+    private static final int POLLING_TOLERANCE_SECONDS = 5;
 
     /**
      * The information supplied by workers the last time they polled for more tasks.
@@ -74,9 +76,9 @@ public synchronized void catalog (WorkerStatus workerStatus) {
      */
     private synchronized void purgeDeadWorkers () {
         long now = System.currentTimeMillis();
-        long oldestAcceptable = now - WORKER_RECORD_DURATION_MSEC;
         List<WorkerObservation> ancientObservations = new ArrayList<>();
         for (WorkerObservation observation : observationsByWorkerId.values()) {
+            long oldestAcceptable = now - ((observation.status.pollIntervalSeconds + POLLING_TOLERANCE_SECONDS) * 1000);
             if (observation.lastSeen < oldestAcceptable) {
                 ancientObservations.add(observation);
             }
diff --git a/src/main/java/com/conveyal/analysis/controllers/BrokerController.java b/src/main/java/com/conveyal/analysis/controllers/BrokerController.java
index 409180fef..9deb1a882 100644
--- a/src/main/java/com/conveyal/analysis/controllers/BrokerController.java
+++ b/src/main/java/com/conveyal/analysis/controllers/BrokerController.java
@@ -14,6 +14,7 @@
 import com.conveyal.analysis.persistence.Persistence;
 import com.conveyal.analysis.util.HttpStatus;
 import com.conveyal.analysis.util.JsonUtil;
+import com.conveyal.gtfs.util.Util;
 import com.conveyal.r5.analyst.WorkerCategory;
 import com.conveyal.r5.analyst.cluster.AnalysisWorker;
 import com.conveyal.r5.analyst.cluster.RegionalTask;
@@ -45,6 +46,7 @@
 
 import java.net.NoRouteToHostException;
 import java.net.SocketTimeoutException;
+import java.nio.charset.StandardCharsets;
 import java.util.ArrayList;
 import java.util.Collection;
 import java.util.HashMap;
@@ -52,6 +54,7 @@
 import java.util.Map;
 import java.util.UUID;
 
+import static com.conveyal.r5.common.Util.human;
 import static com.conveyal.r5.common.Util.notNullOrEmpty;
 import static com.google.common.base.Preconditions.checkNotNull;
 
@@ -289,7 +292,8 @@ private String jsonResponse (Response response, int statusCode, Object object) {
         // We could call response.body(jsonMapper.writeValueAsBytes(object));
         // but then the calling handler functions need to explicitly return null which is weird.
         try {
-            return jsonMapper.writeValueAsString(object);
+            String body = jsonMapper.writeValueAsString(object);
+            return body;
         } catch (JsonProcessingException e) {
             throw AnalysisServerException.unknown(e);
         }
@@ -362,7 +366,7 @@ private Object workerPoll (Request request, Response response) {
         broker.recordWorkerObservation(workerStatus);
         WorkerCategory workerCategory = workerStatus.getWorkerCategory();
         // See if any appropriate tasks exist for this worker.
-        List<RegionalTask> tasks = broker.getSomeWork(workerCategory);
+        List<RegionalTask> tasks = broker.getSomeWork(workerCategory, workerStatus.maxTasksRequested);
         // If there is no work for the worker, signal this clearly with a "no content" code,
         // so the worker can sleep a while before the next polling attempt.
         if (tasks.isEmpty()) {
diff --git a/src/main/java/com/conveyal/analysis/models/AnalysisRequest.java b/src/main/java/com/conveyal/analysis/models/AnalysisRequest.java
index 10d326656..b8859d9d5 100644
--- a/src/main/java/com/conveyal/analysis/models/AnalysisRequest.java
+++ b/src/main/java/com/conveyal/analysis/models/AnalysisRequest.java
@@ -161,6 +161,21 @@ public class AnalysisRequest {
      */
     public ChaosParameters injectFault;
 
+    /**
+     * Whether to include the number of opportunities reached during each minute of travel in results sent back
+     * to the broker. Requires both an origin and destination pointset to be specified, and in the case of regional
+     * analyses the origins must be non-gridded, and results will be collated to CSV.
+     * It should be possible to enable regional results for gridded origins as well.
+     */
+    public boolean includeTemporalDensity = false;
+
+    /**
+     * If this is set to a value above zero, report the amount of time needed to reach the given number of
+     * opportunities from this origin (known technically as "dual accessibility").
+     */
+    public int dualAccessibilityThreshold = 0;
+
+
     /**
      * Create the R5 `Scenario` from this request.
      */
@@ -265,6 +280,9 @@ public void populateTask (AnalysisWorkerTask task, UserPermissions userPermissio
                 throw new IllegalArgumentException("Must be admin user to inject faults.");
             }
         }
+
+        task.includeTemporalDensity = includeTemporalDensity;
+        task.dualAccessibilityThreshold = dualAccessibilityThreshold;
     }
 
     private EnumSet<LegMode> getEnumSetFromString (String s) {
diff --git a/src/main/java/com/conveyal/analysis/models/ConvertToFrequency.java b/src/main/java/com/conveyal/analysis/models/ConvertToFrequency.java
index 0fbbddf46..812e4caca 100644
--- a/src/main/java/com/conveyal/analysis/models/ConvertToFrequency.java
+++ b/src/main/java/com/conveyal/analysis/models/ConvertToFrequency.java
@@ -32,9 +32,6 @@ public static class FrequencyEntry extends AbstractTimetable {
         /** trip from which to copy travel times */
         public String sourceTrip;
 
-        /** trips on the selected patterns which could be used as source trips */
-        public String[] patternTrips;
-
         public AddTrips.PatternTimetable toR5 (String feed) {
             AddTrips.PatternTimetable pt = toBaseR5Timetable();
 
diff --git a/src/main/java/com/conveyal/analysis/results/CsvResultType.java b/src/main/java/com/conveyal/analysis/results/CsvResultType.java
index cdf60e410..29446ba98 100644
--- a/src/main/java/com/conveyal/analysis/results/CsvResultType.java
+++ b/src/main/java/com/conveyal/analysis/results/CsvResultType.java
@@ -5,5 +5,5 @@
  * do serve to enumerate the acceptable parameters coming over the HTTP API.
  */
 public enum CsvResultType {
-    ACCESS, TIMES, PATHS
+    ACCESS, TIMES, PATHS, TDENSITY
 }
diff --git a/src/main/java/com/conveyal/analysis/results/CsvResultWriter.java b/src/main/java/com/conveyal/analysis/results/CsvResultWriter.java
index e07abc2af..018931209 100644
--- a/src/main/java/com/conveyal/analysis/results/CsvResultWriter.java
+++ b/src/main/java/com/conveyal/analysis/results/CsvResultWriter.java
@@ -12,6 +12,8 @@
 import java.io.FileWriter;
 import java.io.IOException;
 
+import static com.google.common.base.Preconditions.checkArgument;
+import static com.google.common.base.Preconditions.checkNotNull;
 import static com.google.common.base.Preconditions.checkState;
 
 /**
@@ -55,6 +57,7 @@ public abstract class CsvResultWriter extends BaseResultWriter implements Region
      */
     CsvResultWriter (RegionalTask task, FileStorage fileStorage) throws IOException {
         super(fileStorage);
+        checkArgument(task.originPointSet != null, "CsvResultWriters require FreeFormPointSet origins.");
         super.prepare(task.jobId);
         this.fileName = task.jobId + "_" + resultType() +".csv";
         BufferedWriter bufferedWriter = new BufferedWriter(new FileWriter(bufferFile));
diff --git a/src/main/java/com/conveyal/analysis/results/MultiOriginAssembler.java b/src/main/java/com/conveyal/analysis/results/MultiOriginAssembler.java
index 3bbd5915f..17fc4902a 100644
--- a/src/main/java/com/conveyal/analysis/results/MultiOriginAssembler.java
+++ b/src/main/java/com/conveyal/analysis/results/MultiOriginAssembler.java
@@ -109,8 +109,10 @@ public MultiOriginAssembler (RegionalAnalysis regionalAnalysis, Job job, FileSto
 
             if (job.templateTask.recordAccessibility) {
                 if (job.templateTask.originPointSet != null) {
+                    // Freeform origins - create CSV regional analysis results
                     resultWriters.add(new AccessCsvResultWriter(job.templateTask, fileStorage));
                 } else {
+                    // Gridded origins - create gridded regional analysis results
                     resultWriters.add(new MultiGridResultWriter(regionalAnalysis, job.templateTask, fileStorage));
                 }
             }
@@ -123,6 +125,20 @@ public MultiOriginAssembler (RegionalAnalysis regionalAnalysis, Job job, FileSto
                 resultWriters.add(new PathCsvResultWriter(job.templateTask, fileStorage));
             }
 
+            if (job.templateTask.includeTemporalDensity) {
+                if (job.templateTask.originPointSet == null) {
+                    // Gridded origins. The full temporal density information is probably too voluminous to be useful.
+                    // We might want to record a grid of dual accessibility values, but this will require some serious
+                    // refactoring of the GridResultWriter.
+                    // if (job.templateTask.dualAccessibilityThreshold > 0) { ... }
+                    throw new IllegalArgumentException("Temporal density of opportunities cannot be recorded for gridded origin points.");
+                } else {
+                    // Freeform origins.
+                    // Output includes temporal density of opportunities and optionally dual accessibility.
+                    resultWriters.add(new TemporalDensityCsvResultWriter(job.templateTask, fileStorage));
+                }
+            }
+
             checkArgument(job.templateTask.makeTauiSite || notNullOrEmpty(resultWriters),
                 "A non-Taui regional analysis should always create at least one grid or CSV file.");
 
diff --git a/src/main/java/com/conveyal/analysis/results/TemporalDensityCsvResultWriter.java b/src/main/java/com/conveyal/analysis/results/TemporalDensityCsvResultWriter.java
new file mode 100644
index 000000000..61279629c
--- /dev/null
+++ b/src/main/java/com/conveyal/analysis/results/TemporalDensityCsvResultWriter.java
@@ -0,0 +1,90 @@
+package com.conveyal.analysis.results;
+
+import com.conveyal.file.FileStorage;
+import com.conveyal.r5.analyst.cluster.RegionalTask;
+import com.conveyal.r5.analyst.cluster.RegionalWorkResult;
+
+import java.io.IOException;
+import java.util.ArrayList;
+import java.util.List;
+
+/**
+ * This handles collating regional results into CSV files containing temporal opportunity density
+ * (number of opportunities reached in each one-minute interval, the derivative of step-function accessibility)
+ * as well as "dual" accessibility (the amount of time needed to reach n opportunities).
+ */
+public class TemporalDensityCsvResultWriter extends CsvResultWriter {
+
+    private final int dualThreshold;
+
+    public TemporalDensityCsvResultWriter(RegionalTask task, FileStorage fileStorage) throws IOException {
+        super(task, fileStorage);
+        dualThreshold = task.dualAccessibilityThreshold;
+    }
+
+    @Override
+    public CsvResultType resultType () {
+        return CsvResultType.TDENSITY;
+    }
+
+    @Override
+    public String[] columnHeaders () {
+        List<String> headers = new ArrayList<>();
+        // The ids of the freeform origin point and destination set
+        headers.add("originId");
+        headers.add("destId");
+        headers.add("percentile");
+        for (int m = 0; m < 120; m += 1) {
+            // The opportunity density over travel minute m
+            headers.add(Integer.toString(m));
+        }
+        // The number of minutes needed to reach d destination opportunities
+        headers.add("D" + dualThreshold);
+        return headers.toArray(new String[0]);
+    }
+
+    @Override
+    protected void checkDimension (RegionalWorkResult workResult) {
+        checkDimension(
+            workResult, "destination pointsets",
+            workResult.opportunitiesPerMinute.length, task.destinationPointSetKeys.length
+        );
+        for (double[][] percentilesForPointset : workResult.opportunitiesPerMinute) {
+            checkDimension(workResult, "percentiles", percentilesForPointset.length, task.percentiles.length);
+            for (double[] minutesForPercentile : percentilesForPointset) {
+                checkDimension(workResult, "minutes", minutesForPercentile.length, 120);
+            }
+        }
+    }
+
+    @Override
+    public Iterable<String[]> rowValues (RegionalWorkResult workResult) {
+        List<String[]> rows = new ArrayList<>();
+        String originId = task.originPointSet.getId(workResult.taskId);
+        for (int d = 0; d < task.destinationPointSetKeys.length; d++) {
+            double[][] percentilesForDestPointset = workResult.opportunitiesPerMinute[d];
+            for (int p = 0; p < task.percentiles.length; p++) {
+                List<String> row = new ArrayList<>(125);
+                row.add(originId);
+                row.add(task.destinationPointSetKeys[d]);
+                row.add(Integer.toString(p));
+                // One density value for each of 120 minutes
+                double[] densitiesPerMinute = percentilesForDestPointset[p];
+                for (int m = 0; m < 120; m++) {
+                    row.add(Double.toString(densitiesPerMinute[m]));
+                }
+                // One dual accessibility value
+                int m = 0;
+                double sum = 0;
+                while (sum < dualThreshold && m < 120) {
+                    sum += densitiesPerMinute[m];
+                    m += 1;
+                }
+                row.add(Integer.toString(m >= 120 ? -1 : m));
+                rows.add(row.toArray(new String[row.size()]));
+            }
+        }
+        return rows;
+    }
+
+}
diff --git a/src/main/java/com/conveyal/gtfs/error/SuspectStopLocationError.java b/src/main/java/com/conveyal/gtfs/error/SuspectStopLocationError.java
new file mode 100644
index 000000000..ac2841eea
--- /dev/null
+++ b/src/main/java/com/conveyal/gtfs/error/SuspectStopLocationError.java
@@ -0,0 +1,31 @@
+package com.conveyal.gtfs.error;
+
+import com.conveyal.gtfs.validator.model.Priority;
+
+import java.io.Serializable;
+
+/**
+ * Indicates that a stop is in a suspect location, for example in a place like a desert where there are not enough
+ * people to support public transit. This can be the result of incorrect coordinate transforms into WGS84.
+ * Stops are often located on "null island" at (0,0). This can also happen in other coordinate systems before they
+ * are transformed to WGS84: the origin of a common French coordinate system is in the Sahara.
+ */
+public class SuspectStopLocationError extends GTFSError implements Serializable {
+    public static final long serialVersionUID = 1L;
+
+    public SuspectStopLocationError(String stopId, long line) {
+        super("stops", line, "stop_id", stopId);
+    }
+
+    @Override public String getMessage() {
+        return String.format(
+            "Stop with ID %s is in a sparsely populated area (fewer than 5 inhabitants per square km in any " +
+                    "neighboring 1/4 degree cell)",
+            affectedEntityId
+        );
+    }
+
+    @Override public Priority getPriority() {
+        return Priority.MEDIUM;
+    }
+}
diff --git a/src/main/java/com/conveyal/gtfs/model/Agency.java b/src/main/java/com/conveyal/gtfs/model/Agency.java
index fd1e6031c..32c221ea4 100644
--- a/src/main/java/com/conveyal/gtfs/model/Agency.java
+++ b/src/main/java/com/conveyal/gtfs/model/Agency.java
@@ -39,7 +39,7 @@ protected boolean isRequired() {
         @Override
         public void loadOneRow() throws IOException {
             Agency a = new Agency();
-            a.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            a.sourceFileLine = row;
             a.agency_id    = getStringField("agency_id", false); // can only be absent if there is a single agency -- requires a special validator.
             a.agency_name  = getStringField("agency_name", true);
             a.agency_url   = getUrlField("agency_url", true);
diff --git a/src/main/java/com/conveyal/gtfs/model/Calendar.java b/src/main/java/com/conveyal/gtfs/model/Calendar.java
index 8ac72a000..74db9c8f8 100644
--- a/src/main/java/com/conveyal/gtfs/model/Calendar.java
+++ b/src/main/java/com/conveyal/gtfs/model/Calendar.java
@@ -55,7 +55,7 @@ public void loadOneRow() throws IOException {
                 feed.errors.add(new DuplicateKeyError(tableName, row, "service_id", service_id));
             } else {
                 Calendar c = new Calendar();
-                c.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+                c.sourceFileLine = row;
                 c.service_id = service.service_id;
                 c.monday = getIntField("monday", true, 0, 1);
                 c.tuesday = getIntField("tuesday", true, 0, 1);
diff --git a/src/main/java/com/conveyal/gtfs/model/CalendarDate.java b/src/main/java/com/conveyal/gtfs/model/CalendarDate.java
index db4a1e9f3..cd8b36053 100644
--- a/src/main/java/com/conveyal/gtfs/model/CalendarDate.java
+++ b/src/main/java/com/conveyal/gtfs/model/CalendarDate.java
@@ -56,7 +56,7 @@ public void loadOneRow() throws IOException {
                 feed.errors.add(new DuplicateKeyError(tableName, row, "(service_id, date)", keyString));
             } else {
                 CalendarDate cd = new CalendarDate();
-                cd.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+                cd.sourceFileLine = row;
                 cd.service_id = service_id;
                 cd.date = date;
                 cd.exception_type = getIntField("exception_type", true, 1, 2);
diff --git a/src/main/java/com/conveyal/gtfs/model/Entity.java b/src/main/java/com/conveyal/gtfs/model/Entity.java
index a2395b473..16431860b 100644
--- a/src/main/java/com/conveyal/gtfs/model/Entity.java
+++ b/src/main/java/com/conveyal/gtfs/model/Entity.java
@@ -85,7 +85,11 @@ public static abstract class Loader<E extends Entity> {
         protected final Set<String> missingRequiredColumns = Sets.newHashSet();
 
         protected CsvReader reader;
-        protected int       row;
+
+        // Use one-based rows which are easier to understand in error messages.
+        // Line numbers are conventionally one-based even in IDEs and programming editors.
+        protected int row = 1;
+
         // TODO "String column" that is set before any calls to avoid passing around the column name
 
         public Loader(GTFSFeed feed, String tableName) {
@@ -289,18 +293,22 @@ public void loadTable (ZipFile zip) throws IOException {
             }
             CsvReader reader = new CsvReader(inStream, ',', Charset.forName("UTF8"));
             this.reader = reader;
+            // row number is already positioned at 1
             boolean hasHeaders = reader.readHeaders();
             if (!hasHeaders) {
                 feed.errors.add(new EmptyTableError(tableName));
             }
             while (reader.readRecord()) {
-                // reader.getCurrentRecord() is zero-based and does not include the header line, keep our own row count
-                if (++row % 500000 == 0) {
+                // reader.getCurrentRecord() is zero-based and does not include the header line,
+                // so we keep our own one-based row count that does include the header line.
+                // Advance row number before calling loadOneRow so the current value is available inside that function.
+                row += 1;
+                if (row % 500000 == 0) {
                     LOG.info("Record number {}", human(row));
                 }
                 loadOneRow(); // Call subclass method to produce an entity from the current row.
             }
-            if (row == 0) {
+            if (row < 2) {
                 feed.errors.add(new EmptyTableError(tableName));
             }
         }
diff --git a/src/main/java/com/conveyal/gtfs/model/FareAttribute.java b/src/main/java/com/conveyal/gtfs/model/FareAttribute.java
index 9ea58bede..3803920e2 100644
--- a/src/main/java/com/conveyal/gtfs/model/FareAttribute.java
+++ b/src/main/java/com/conveyal/gtfs/model/FareAttribute.java
@@ -42,7 +42,7 @@ public void loadOneRow() throws IOException {
                 feed.errors.add(new DuplicateKeyError(tableName, row, "fare_id", fareId));
             } else {
                 FareAttribute fa = new FareAttribute();
-                fa.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+                fa.sourceFileLine = row;
                 fa.fare_id = fareId;
                 fa.price = getDoubleField("price", true, 0, Integer.MAX_VALUE);
                 fa.currency_type = getStringField("currency_type", true);
diff --git a/src/main/java/com/conveyal/gtfs/model/FareRule.java b/src/main/java/com/conveyal/gtfs/model/FareRule.java
index ce8b3c033..04cfdb7bb 100644
--- a/src/main/java/com/conveyal/gtfs/model/FareRule.java
+++ b/src/main/java/com/conveyal/gtfs/model/FareRule.java
@@ -43,7 +43,7 @@ public void loadOneRow() throws IOException {
 
             Fare fare = fares.computeIfAbsent(fareId, Fare::new);
             FareRule fr = new FareRule();
-            fr.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            fr.sourceFileLine = row;
             fr.fare_id = fare.fare_id;
             fr.route_id = getStringField("route_id", false);
             fr.origin_id = getStringField("origin_id", false);
diff --git a/src/main/java/com/conveyal/gtfs/model/FeedInfo.java b/src/main/java/com/conveyal/gtfs/model/FeedInfo.java
index cfedca049..041a2f943 100644
--- a/src/main/java/com/conveyal/gtfs/model/FeedInfo.java
+++ b/src/main/java/com/conveyal/gtfs/model/FeedInfo.java
@@ -41,7 +41,7 @@ protected boolean isRequired() {
         @Override
         public void loadOneRow() throws IOException {
             FeedInfo fi = new FeedInfo();
-            fi.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            fi.sourceFileLine = row;
             fi.feed_id = getStringField("feed_id", false);
             fi.feed_publisher_name = getStringField("feed_publisher_name", true);
             fi.feed_publisher_url = getUrlField("feed_publisher_url", true);
diff --git a/src/main/java/com/conveyal/gtfs/model/Frequency.java b/src/main/java/com/conveyal/gtfs/model/Frequency.java
index e327224aa..558d7bc41 100644
--- a/src/main/java/com/conveyal/gtfs/model/Frequency.java
+++ b/src/main/java/com/conveyal/gtfs/model/Frequency.java
@@ -57,7 +57,7 @@ protected boolean isRequired() {
         public void loadOneRow() throws IOException {
             Frequency f = new Frequency();
             Trip trip = getRefField("trip_id", true, feed.trips);
-            f.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            f.sourceFileLine = row;
             f.trip_id = trip.trip_id;
             f.start_time = getTimeField("start_time", true);
             f.end_time = getTimeField("end_time", true);
diff --git a/src/main/java/com/conveyal/gtfs/model/Route.java b/src/main/java/com/conveyal/gtfs/model/Route.java
index f4965861b..d6faf1fcf 100644
--- a/src/main/java/com/conveyal/gtfs/model/Route.java
+++ b/src/main/java/com/conveyal/gtfs/model/Route.java
@@ -51,7 +51,7 @@ protected boolean isRequired() {
         @Override
         public void loadOneRow() throws IOException {
             Route r = new Route();
-            r.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            r.sourceFileLine = row;
             r.route_id = getStringField("route_id", true);
             Agency agency = getRefField("agency_id", false, feed.agency);
 
diff --git a/src/main/java/com/conveyal/gtfs/model/ShapePoint.java b/src/main/java/com/conveyal/gtfs/model/ShapePoint.java
index 904cb939b..e5a7e71d1 100644
--- a/src/main/java/com/conveyal/gtfs/model/ShapePoint.java
+++ b/src/main/java/com/conveyal/gtfs/model/ShapePoint.java
@@ -46,7 +46,7 @@ public void loadOneRow() throws IOException {
             double shape_dist_traveled = getDoubleField("shape_dist_traveled", false, 0D, Double.MAX_VALUE);
 
             ShapePoint s = new ShapePoint(shape_id, shape_pt_lat, shape_pt_lon, shape_pt_sequence, shape_dist_traveled);
-            s.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            s.sourceFileLine = row;
             feed.shape_points.put(new Tuple2<String, Integer>(s.shape_id, s.shape_pt_sequence), s);
         }
     }
diff --git a/src/main/java/com/conveyal/gtfs/model/Stop.java b/src/main/java/com/conveyal/gtfs/model/Stop.java
index 26a5fabb3..16790051b 100644
--- a/src/main/java/com/conveyal/gtfs/model/Stop.java
+++ b/src/main/java/com/conveyal/gtfs/model/Stop.java
@@ -45,7 +45,7 @@ protected boolean isRequired() {
         @Override
         public void loadOneRow() throws IOException {
             Stop s = new Stop();
-            s.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            s.sourceFileLine = row;
             s.stop_id   = getStringField("stop_id", true);
             s.stop_code = getStringField("stop_code", false);
             s.stop_name = getStringField("stop_name", true);
diff --git a/src/main/java/com/conveyal/gtfs/model/StopTime.java b/src/main/java/com/conveyal/gtfs/model/StopTime.java
index daabfec14..31d5d9159 100644
--- a/src/main/java/com/conveyal/gtfs/model/StopTime.java
+++ b/src/main/java/com/conveyal/gtfs/model/StopTime.java
@@ -50,7 +50,7 @@ protected boolean isRequired() {
         @Override
         public void loadOneRow() throws IOException {
             StopTime st = new StopTime();
-            st.sourceFileLine = row + 1; // offset line number by 1 to account for 0-based row index
+            st.sourceFileLine = row;
             st.trip_id        = getStringField("trip_id", true);
             // TODO: arrival_time and departure time are not required, but if one is present the other should be
             // also, if this is the first or last stop, they are both required
diff --git a/src/main/java/com/conveyal/gtfs/model/Transfer.java b/src/main/java/com/conveyal/gtfs/model/Transfer.java
index 56066c373..3af7b970b 100644
--- a/src/main/java/com/conveyal/gtfs/model/Transfer.java
+++ b/src/main/java/com/conveyal/gtfs/model/Transfer.java
@@ -31,7 +31,7 @@ protected boolean isRequired() {
         @Override
         public void loadOneRow() throws IOException {
             Transfer tr = new Transfer();
-            tr.sourceFileLine    = row + 1; // offset line number by 1 to account for 0-based row index
+            tr.sourceFileLine    = row;
             tr.from_stop_id      = getStringField("from_stop_id", true);
             tr.to_stop_id        = getStringField("to_stop_id", true);
             tr.transfer_type     = getIntField("transfer_type", true, 0, 3);
@@ -48,6 +48,7 @@ public void loadOneRow() throws IOException {
             getRefField("from_trip_id", false, feed.trips);
             getRefField("to_trip_id", false, feed.trips);
 
+            // row number used as an arbitrary unique string to give MapDB a key.
             feed.transfers.put(Long.toString(row), tr);
         }
 
diff --git a/src/main/java/com/conveyal/gtfs/model/Trip.java b/src/main/java/com/conveyal/gtfs/model/Trip.java
index d0a16380e..f1f85e184 100644
--- a/src/main/java/com/conveyal/gtfs/model/Trip.java
+++ b/src/main/java/com/conveyal/gtfs/model/Trip.java
@@ -40,7 +40,7 @@ protected boolean isRequired() {
         public void loadOneRow() throws IOException {
             Trip t = new Trip();
 
-            t.sourceFileLine  = row + 1; // offset line number by 1 to account for 0-based row index
+            t.sourceFileLine  = row;
             t.route_id        = getStringField("route_id", true);
             t.service_id      = getStringField("service_id", true);
             t.trip_id         = getStringField("trip_id", true);
diff --git a/src/main/java/com/conveyal/gtfs/storage/BooleanAsciiGrid.java b/src/main/java/com/conveyal/gtfs/storage/BooleanAsciiGrid.java
index 91293671e..f7f0c8045 100644
--- a/src/main/java/com/conveyal/gtfs/storage/BooleanAsciiGrid.java
+++ b/src/main/java/com/conveyal/gtfs/storage/BooleanAsciiGrid.java
@@ -9,7 +9,7 @@
 
 /**
  * Loads an ESRI ASCII grid containing integers and allows looking up values as booleans (where > 0).
- * This is used for
+ * This can be used for a basic stop location test using a simple population raster.
  */
 public class BooleanAsciiGrid {
 
@@ -51,19 +51,22 @@ public BooleanAsciiGrid (InputStream inputStream, boolean gzipped) {
     }
 
     /**
-     * Get a grid for places where population density is over 5 people per square kilometer.
+     * Get a grid for places where population density is over 5 people per square kilometer in any neighboring cell.
      * We use the Gridded Population of the World v3 data set for 2015 UN-adjusted population density.
      * This data set was downloaded at 1/4 degree resolution in ESRI ASCII grid format. The grid file was edited
      * manually to eliminate the no-data value header, since I could not find a way to operate on no-value cells in the
      * QGIS raster calculator. Then the raster calculator in QGIS was used with the formula ("glds00ag15@1" > 5),
      * which makes all cells with population density above the threshold have a value of one,
-     * and all others a value of zero (since the no data value in the grid is -9999). This was then exported as another
-     * ASCII grid file, which zips well. The license for this data set is Creative Commons Attribution.
+     * and all others a value of zero (since the no data value in the grid is -9999). Next, the GRASS r.neighbors tool
+     * in QGIS was used to blur this layer, adding to each cell the sum of the neighboring eight cells ("neighborhood
+     * size" of 3, the dimension in each direction). A final logical operation ("blurred@1" > 0) was applied, with the
+     * result exported as another ASCII grid file, which zips well. The license for this data set is Creative Commons
+     * Attribution.
      * See http://sedac.ciesin.columbia.edu/data/collection/gpw-v3
      */
     public static BooleanAsciiGrid forEarthPopulation() {
         try {
-            InputStream gridStream = BooleanAsciiGrid.class.getResourceAsStream("gpwv3-quarter-boolean.asc");
+            InputStream gridStream = BooleanAsciiGrid.class.getResourceAsStream("gpwv3-quarter-buffer-boolean.asc");
             return new BooleanAsciiGrid(gridStream, false);
         } catch (Exception ex) {
             throw new RuntimeException(ex);
diff --git a/src/main/java/com/conveyal/gtfs/validator/PostLoadValidator.java b/src/main/java/com/conveyal/gtfs/validator/PostLoadValidator.java
index 303bc3537..a4544703a 100644
--- a/src/main/java/com/conveyal/gtfs/validator/PostLoadValidator.java
+++ b/src/main/java/com/conveyal/gtfs/validator/PostLoadValidator.java
@@ -6,7 +6,9 @@
 import com.conveyal.gtfs.error.GeneralError;
 import com.conveyal.gtfs.error.RangeError;
 import com.conveyal.gtfs.error.ReferentialIntegrityError;
+import com.conveyal.gtfs.error.SuspectStopLocationError;
 import com.conveyal.gtfs.model.Stop;
+import com.conveyal.gtfs.storage.BooleanAsciiGrid;
 
 import java.util.List;
 
@@ -33,6 +35,7 @@ public PostLoadValidator (GTFSFeed feed) {
     public void validate () {
         validateCalendarServices();
         validateParentStations();
+        validateStopPopulationDensity();
     }
 
     /**
@@ -46,6 +49,20 @@ private void validateCalendarServices () {
         }
     }
 
+    /**
+     * Validate that stops are not in locations with no people. This can happen from incorrect coordinate transforms
+     * into WGS84. Stops are often located on "null island" at (0,0). This can also happen in other coordinate systems
+     * before they are transformed to WGS84: the origin of a common French coordinate system is in the Sahara.
+     */
+    private void validateStopPopulationDensity () {
+        BooleanAsciiGrid popGrid = BooleanAsciiGrid.forEarthPopulation();
+        for (Stop stop : feed.stops.values()) {
+            if (!(popGrid.getValueForCoords(stop.stop_lon, stop.stop_lat))) {
+                feed.errors.add(new SuspectStopLocationError(stop.stop_id, stop.sourceFileLine));
+            }
+        }
+    }
+
     /**
      * Validate location_type and parent_station constraints as well as referential integrity.
      * Individual validation actions like this could be factored out into separate classes (PostLoadValidators)
diff --git a/src/main/java/com/conveyal/r5/OneOriginResult.java b/src/main/java/com/conveyal/r5/OneOriginResult.java
index 6c721d41e..090cdd9a0 100644
--- a/src/main/java/com/conveyal/r5/OneOriginResult.java
+++ b/src/main/java/com/conveyal/r5/OneOriginResult.java
@@ -1,6 +1,7 @@
 package com.conveyal.r5;
 
 import com.conveyal.r5.analyst.AccessibilityResult;
+import com.conveyal.r5.analyst.TemporalDensityResult;
 import com.conveyal.r5.analyst.cluster.PathResult;
 import com.conveyal.r5.analyst.cluster.TravelTimeResult;
 
@@ -20,10 +21,18 @@ public class OneOriginResult {
 
     public final PathResult paths;
 
-    public OneOriginResult(TravelTimeResult travelTimes, AccessibilityResult accessibility, PathResult paths) {
+    public final TemporalDensityResult density;
+
+    public OneOriginResult(
+            TravelTimeResult travelTimes,
+            AccessibilityResult accessibility,
+            PathResult paths,
+            TemporalDensityResult density
+    ) {
         this.travelTimes = travelTimes;
         this.accessibility = accessibility;
         this.paths = paths;
+        this.density = density;
     }
 
 }
diff --git a/src/main/java/com/conveyal/r5/analyst/TemporalDensityResult.java b/src/main/java/com/conveyal/r5/analyst/TemporalDensityResult.java
new file mode 100644
index 000000000..60956048c
--- /dev/null
+++ b/src/main/java/com/conveyal/r5/analyst/TemporalDensityResult.java
@@ -0,0 +1,95 @@
+package com.conveyal.r5.analyst;
+
+import com.conveyal.r5.analyst.cluster.AnalysisWorkerTask;
+import com.google.common.base.Preconditions;
+
+import static com.conveyal.r5.common.Util.notNullOrEmpty;
+import static com.conveyal.r5.profile.FastRaptorWorker.UNREACHED;
+
+/**
+ * An instance of this is included in a OneOriginResult for reporting how many opportunities are encountered during each
+ * minute of travel. If we use more than one destination point set they are already constrained to all be aligned with
+ * the same number of destinations.
+ *
+ * The data retained here feed into three different kinds of results: "Dual" accessibility (the number of opportunities
+ * reached in a given number of minutes of travel time); temporal opportunity density (analogous to a probability density
+ * function, how many opportunities are encountered during each minute of travel, whose integral is the cumulative
+ * accessibility curve).
+ *
+ * Originally this class was tracking the identity of the N nearest points rather than just binning them by travel time.
+ * This is more efficient in cases where N is small, and allows retaining the one-second resolution. However currently
+ * there does not seem to be much demand among users for this level of detail, so it has been removed in the interest
+ * of simplicity and maintainability. See issue 884 for more comments on implementation trade-offs.
+ */
+public class TemporalDensityResult {
+
+    // Internal state fields
+
+    private final PointSet[] destinationPointSets;
+    private final int nPercentiles;
+    private final int opportunityThreshold;
+
+    // Externally visible fields for accumulating results
+
+    /**
+     * The temporal density of opportunities. For each destination set, for each percentile, for each minute of
+     * travel from 0 to 120, the number of opportunities reached in travel times from i (inclusive) to i+1 (exclusive).
+     */
+    public final double[][][] opportunitiesPerMinute;
+
+    public TemporalDensityResult(AnalysisWorkerTask task) {
+        Preconditions.checkArgument(
+            notNullOrEmpty(task.destinationPointSets),
+            "Dual accessibility requires at least one destination pointset."
+        );
+        this.destinationPointSets = task.destinationPointSets;
+        this.nPercentiles = task.percentiles.length;
+        this.opportunityThreshold = task.dualAccessibilityThreshold;
+        this.opportunitiesPerMinute = new double[destinationPointSets.length][nPercentiles][120];
+    }
+
+    public void recordOneTarget (int target, int[] travelTimePercentilesSeconds) {
+        // Increment histogram bin for the number of minutes of travel by the number of opportunities at the target.
+        for (int d = 0; d < destinationPointSets.length; d++) {
+            PointSet dps = destinationPointSets[d];
+            for (int p = 0; p < nPercentiles; p++) {
+                if (travelTimePercentilesSeconds[p] == UNREACHED) {
+                    break; // If any percentile is unreached, all higher ones are also unreached.
+                }
+                int m = travelTimePercentilesSeconds[p] / 60;
+                if (m <= 120) {
+                    opportunitiesPerMinute[d][p][m] += dps.getOpportunityCount(target);
+                }
+            }
+        }
+    }
+
+    /**
+     * Calculate "dual" accessibility from the accumulated temporal opportunity density array.
+     * @param n the threshold quantity of opportunities
+     * @return the minimum whole number of minutes necessary to reach n opportunities,
+     *         for each destination set and percentile of travel time.
+     */
+    public int[][] minutesToReachOpportunities(int n) {
+        int[][] result = new int[destinationPointSets.length][nPercentiles];
+        for (int d = 0; d < destinationPointSets.length; d++) {
+            for (int p = 0; p < nPercentiles; p++) {
+                result[d][p] = -1;
+                double count = 0;
+                for (int m = 0; m < 120; m++) {
+                    count += opportunitiesPerMinute[d][p][m];
+                    if (count >= n) {
+                        result[d][p] = m + 1;
+                        break;
+                    }
+                }
+            }
+        }
+        return result;
+    }
+
+    public int[][] minutesToReachOpportunities() {
+        return minutesToReachOpportunities(opportunityThreshold);
+    }
+
+}
diff --git a/src/main/java/com/conveyal/r5/analyst/TravelTimeReducer.java b/src/main/java/com/conveyal/r5/analyst/TravelTimeReducer.java
index 4e1ca30bc..15ce02012 100644
--- a/src/main/java/com/conveyal/r5/analyst/TravelTimeReducer.java
+++ b/src/main/java/com/conveyal/r5/analyst/TravelTimeReducer.java
@@ -46,6 +46,9 @@ public class TravelTimeReducer {
     /** Retains the paths to one or all destinations, for recording in CSV or reporting in the UI. */
     private PathResult pathResult = null;
 
+    /** Represents how many destinations are reached in each minute of travel from this origin. */
+    private TemporalDensityResult temporalDensityResult = null;
+
     /** If we are calculating accessibility, the PointSets containing opportunities. */
     private PointSet[] destinationPointSets;
 
@@ -121,6 +124,8 @@ public TravelTimeReducer (AnalysisWorkerTask task, TransportNetwork network) {
         this.destinationPointSets = task.destinationPointSets;
         if (task instanceof TravelTimeSurfaceTask) {
             calculateTravelTimes = true;
+            // In single-point analyses, destination pointsets may be missing if the user has not selected one in the
+            // UI, or if the user has selected the step decay function instead of one of the other decay functions.
             calculateAccessibility = notNullOrEmpty(task.destinationPointSets);
         } else {
             // Maybe we should define recordAccessibility and recordTimes on the common superclass AnalysisWorkerTask.
@@ -140,6 +145,9 @@ public TravelTimeReducer (AnalysisWorkerTask task, TransportNetwork network) {
         if (task.includePathResults) {
             pathResult = new PathResult(task, network.transitLayer);
         }
+        if (task.includeTemporalDensity) {
+            temporalDensityResult = new TemporalDensityResult(task);
+        }
 
         // Validate and copy the travel time cutoffs, converting them to seconds to avoid repeated multiplication
         // in tight loops. Also find the points where the decay function reaches zero for these cutoffs.
@@ -271,6 +279,9 @@ private void recordTravelTimePercentilesForTarget (int target, int[] travelTimeP
                 }
             }
         }
+        if (temporalDensityResult != null) {
+            temporalDensityResult.recordOneTarget(target, travelTimePercentilesSeconds);
+        }
     }
 
     /**
@@ -329,7 +340,7 @@ private int convertToMinutes (int timeSeconds) {
      * origin point is not connected to the street network.
      */
     public OneOriginResult finish () {
-        return new OneOriginResult(travelTimeResult, accessibilityResult, pathResult);
+        return new OneOriginResult(travelTimeResult, accessibilityResult, pathResult, temporalDensityResult);
     }
 
     /**
diff --git a/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorker.java b/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorker.java
index aad7da99d..2add61702 100644
--- a/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorker.java
+++ b/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorker.java
@@ -13,6 +13,7 @@
 import com.conveyal.r5.analyst.TravelTimeComputer;
 import com.conveyal.r5.analyst.error.TaskError;
 import com.conveyal.r5.common.JsonUtilities;
+import com.conveyal.r5.transit.TransitLayer;
 import com.conveyal.r5.transit.TransportNetwork;
 import com.conveyal.r5.transit.TransportNetworkCache;
 import com.conveyal.r5.transitive.TransitiveNetwork;
@@ -75,8 +76,10 @@ public interface Config {
 
     private static final Logger LOG = LoggerFactory.getLogger(AnalysisWorker.class);
 
-    public static final int POLL_WAIT_SECONDS = 15;
-    public static final int POLL_MAX_RANDOM_WAIT = 5;
+    private static final int POLL_INTERVAL_MIN_SECONDS = 1;
+    private static final int POLL_INTERVAL_MAX_SECONDS = 15;
+    private static final int POLL_JITTER_SECONDS = 5;
+    private static final int QUEUE_SLOTS_PER_PROCESSOR = 8;
 
     /**
      * This timeout should be longer than the longest expected worker calculation for a single-point request.
@@ -111,8 +114,6 @@ public interface Config {
     /** Keeps some TransportNetworks around, lazy-loading or lazy-building them. */
     public final NetworkPreloader networkPreloader;
 
-    private final Random random = new Random();
-
     /** The common root of all API URLs contacted by this worker, e.g. http://localhost:7070/api/ */
     protected final String brokerBaseUrl;
 
@@ -126,8 +127,11 @@ public interface Config {
      */
     private final List<RegionalWorkResult> workResults = new ArrayList<>();
 
-    /** The last time (in milliseconds since the epoch) that we polled for work. */
-    private long lastPollingTime;
+    /**
+     * The last time (in milliseconds since the epoch) that we polled for work.
+     * The initial value of zero causes the worker to poll the backend immediately on startup avoiding a delay.
+     */
+    private long lastPollingTime = 0;
 
     /** Keep track of how many tasks per minute this worker is processing, broken down by scenario ID. */
     private final ThroughputTracker throughputTracker = new ThroughputTracker();
@@ -162,7 +166,7 @@ public static HttpClient makeHttpClient () {
 
     /**
      * A queue to hold a backlog of regional analysis tasks.
-     * This avoids "slow joiner" syndrome where we wait to poll for more work until all N fetched tasks have finished,
+     * This avoids "slow joiner" syndrome where we wait until all N fetched tasks have finished,
      * but one of the tasks takes much longer than all the rest.
      * This should be long enough to hold all that have come in - we don't need to block on polling the manager.
      * Can this be replaced with the general purpose TaskScheduler component?
@@ -199,66 +203,100 @@ public AnalysisWorker (
     /** The main worker event loop which fetches tasks from a broker and schedules them for execution. */
     public void startPolling () {
 
-        // Create executors with up to one thread per processor.
-        // The default task rejection policy is "Abort".
-        // The executor's queue is rather long because some tasks complete very fast and we poll max once per second.
+        // Create an executor with one thread per processor. The default task rejection policy is "Abort".
+        // The number of threads will only increase from the core pool size toward the max pool size when the queue is
+        // full. We no longer exceed the queue length in normal operation, so the thread pool will remain at core size.
+        // "[The] core pool size is the threshold beyond which [an] executor service prefers to queue up the task
+        // [rather] than spawn a new thread." https://stackoverflow.com/a/72684387/778449
+        // The executor's queue is rather long because some tasks complete very fast and we poll at most once per second.
         int availableProcessors = Runtime.getRuntime().availableProcessors();
-        LOG.debug("Java reports the number of available processors is: {}", availableProcessors);
-        int maxThreads = availableProcessors;
-        int taskQueueLength = availableProcessors * 6;
-        LOG.debug("Maximum number of regional processing threads is {}, length of task queue is {}.", maxThreads, taskQueueLength);
+        LOG.info("Java reports the number of available processors is: {}", availableProcessors);
+        final int maxThreads = availableProcessors;
+        final int taskQueueLength = availableProcessors * QUEUE_SLOTS_PER_PROCESSOR;
+        LOG.info("Maximum number of regional processing threads is {}, target length of task queue is {}.", maxThreads, taskQueueLength);
         BlockingQueue<Runnable> taskQueue = new LinkedBlockingQueue<>(taskQueueLength);
-        regionalTaskExecutor = new ThreadPoolExecutor(1, maxThreads, 60, TimeUnit.SECONDS, taskQueue);
+        regionalTaskExecutor = new ThreadPoolExecutor(maxThreads, maxThreads, 60, TimeUnit.SECONDS, taskQueue);
 
-        // Main polling loop to fill the regional work queue.
+        // This is the main polling loop that fills the regional work queue.
         // Go into an endless loop polling for regional tasks that can be computed asynchronously.
         // You'd think the ThreadPoolExecutor could just block when the blocking queue is full, but apparently
         // people all over the world have been jumping through hoops to try to achieve this simple behavior
         // with no real success, at least without writing bug and deadlock-prone custom executor services.
-        // Two alternative approaches are trying to keep the queue full and waiting for the queue to be almost empty.
-        // To keep the queue full, we repeatedly try to add each task to the queue, pausing and retrying when
-        // it's full. To wait until it's almost empty, we could use wait() in a loop and notify() as tasks are handled.
-        // see https://stackoverflow.com/a/15185004/778449
-        // A simpler approach might be to spin-wait checking whether the queue is low and sleeping briefly,
-        // then fetch more work only when the queue is getting empty.
+        // Our current (revised) approach is to slowly spin-wait, checking whether the queue is low and sleeping briefly,
+        // then fetching more work only when the queue is getting empty.
+
+        // Allows slowing down polling when not actively receiving tasks.
+        boolean receivedWorkLastTime = false;
+
+        // Before first polling the broker, randomly wait a few seconds to spread load when many workers start at once.
+        sleepSeconds((new Random()).nextInt(POLL_JITTER_SECONDS));
         while (true) {
-            List<RegionalTask> tasks = getSomeWork();
-            if (tasks == null || tasks.isEmpty()) {
-                // Either there was no work, or some kind of error occurred.
-                // Sleep for a while before polling again, adding a random component to spread out the polling load.
-                // TODO only randomize delay on the first round, after that it's excessive.
-                int randomWait = random.nextInt(POLL_MAX_RANDOM_WAIT);
-                LOG.debug("Polling the broker did not yield any regional tasks. Sleeping {} + {} sec.", POLL_WAIT_SECONDS, randomWait);
-                sleepSeconds(POLL_WAIT_SECONDS + randomWait);
+            // We establish a lower limit on the wait time between polling to avoid flooding the broker with requests.
+            // If worker handles all tasks in its internal queue in less than this time, this is a speed bottleneck.
+            // This can happen in areas unconnected to transit and when travel time cutoffs are very low.
+            sleepSeconds(POLL_INTERVAL_MIN_SECONDS);
+            // Determine whether to poll this cycle - is the queue running empty, or has the maximum interval passed?
+            {
+                long currentTime = System.currentTimeMillis();
+                boolean maxIntervalExceeded = (currentTime - lastPollingTime) > (POLL_INTERVAL_MAX_SECONDS * 1000);
+                int tasksInQueue = taskQueue.size();
+                // Poll any time we have less tasks in the queue than processors.
+                boolean shouldPoll = maxIntervalExceeded || (receivedWorkLastTime && (tasksInQueue < availableProcessors));
+                LOG.debug("Last polled {} sec ago. Task queue length is {}.", (currentTime - lastPollingTime)/1000, taskQueue.size());
+                if (!shouldPoll) {
+                    continue;
+                }
+                lastPollingTime = currentTime;
+            }
+            // This will request tasks even when queue is rather full.
+            // For now, assume more but smaller task and result chunks is better at leveling broker load.
+            int tasksToRequest = taskQueue.remainingCapacity();
+            // Alternatively: Only request tasks when queue is short. Otherwise, only report results and status.
+            // int tasksToRequest = (tasksInQueue < minQueueLength) ? taskQueue.remainingCapacity() : 0;
+
+            List<RegionalTask> tasks = getSomeWork(tasksToRequest);
+            boolean noWorkReceived = tasks == null || tasks.isEmpty();
+            receivedWorkLastTime = !noWorkReceived; // Allows variable speed polling on the next iteration.
+            if (noWorkReceived) {
+                // Either the broker supplied no work or an error occurred.
                 continue;
             }
             for (RegionalTask task : tasks) {
-                // Try to enqueue each task for execution, repeatedly failing until the queue is not full.
-                // The list of fetched tasks essentially serves as a secondary queue, which is awkward. This is using
-                // exceptions for normal flow control, which is nasty. We should do this differently (#596).
-                while (true) {
-                    try {
-                        // TODO define non-anonymous runnable class to instantiate here, specifically for async regional tasks.
-                        regionalTaskExecutor.execute(() -> {
-                            try {
-                                this.handleOneRegionalTask(task);
-                            } catch (Throwable t) {
-                                LOG.error(
-                                    "An error occurred while handling a regional task, reporting to backend. {}",
-                                    ExceptionUtils.stackTraceString(t)
-                                );
-                                synchronized (workResults) {
-                                    workResults.add(new RegionalWorkResult(t, task));
-                                }
-                            }
-                        });
-                        break;
-                    } catch (RejectedExecutionException e) {
-                        // Queue is full, wait a bit and try to feed it more tasks. If worker handles all tasks in its
-                        // internal queue in less than 1 second, this is a speed bottleneck. This happens with regions
-                        // unconnected to transit and with very small travel time cutoffs.
-                        sleepSeconds(1);
-                    }
+                // Executor services require blocking queues of fixed length. Tasks must be enqueued one by one, and
+                // may fail with a RejectedExecutionException if we exceed the queue length. We choose queue length
+                // and requested number of tasks carefully to avoid overfilling the queue, but should handle the
+                // exceptions just in case something is misconfigured.
+                try {
+                    regionalTaskExecutor.execute(new RegionalTaskRunnable(task));
+                } catch (RejectedExecutionException e) {
+                    LOG.error("Regional task could not be enqueued for processing (queue length exceeded). Task dropped.");
+                }
+            }
+        }
+    }
+
+    /**
+     * Runnable inner class which can access the needed methods on AnalysisWorker (handleOneRegionalTask).
+     * However that method is only called from these runnables - it could potentially be inlined into run().
+     */
+    protected class RegionalTaskRunnable implements Runnable {
+        RegionalTask task;
+
+        public RegionalTaskRunnable(RegionalTask task) {
+            this.task = task;
+        }
+
+        @Override
+        public void run() {
+            try {
+                handleOneRegionalTask(task);
+            } catch (Throwable t) {
+                LOG.error(
+                        "An error occurred while handling a regional task, reporting to backend. {}",
+                        ExceptionUtils.stackTraceString(t)
+                );
+                synchronized (workResults) {
+                    workResults.add(new RegionalWorkResult(t, task));
                 }
             }
         }
@@ -344,10 +382,10 @@ private byte[] singlePointResultToBinary (
             timeGridWriter.writeToDataOutput(new LittleEndianDataOutputStream(byteArrayOutputStream));
             addJsonToGrid(
                     byteArrayOutputStream,
-                    oneOriginResult.accessibility,
+                    oneOriginResult,
                     transportNetwork.scenarioApplicationWarnings,
                     transportNetwork.scenarioApplicationInfo,
-                    oneOriginResult.paths != null ? new PathResultSummary(oneOriginResult.paths, transportNetwork.transitLayer) : null
+                    transportNetwork.transitLayer
             );
         }
         // Single-point tasks don't have a job ID. For now, we'll categorize them by scenario ID.
@@ -457,7 +495,7 @@ protected void handleOneRegionalTask (RegionalTask task) throws Throwable {
             // progress. This avoids crashing the backend by sending back massive 2 million element travel times
             // that have already been written to S3, and throwing exceptions on old backends that can't deal with
             // null AccessibilityResults.
-            oneOriginResult = new OneOriginResult(null, new AccessibilityResult(task), null);
+            oneOriginResult = new OneOriginResult(null, new AccessibilityResult(task), null, null);
         }
 
         // Accumulate accessibility results, which will be returned to the backend in batches.
@@ -489,6 +527,8 @@ public static class GridJsonBlock {
 
         public PathResultSummary pathSummaries;
 
+        public double[][][] opportunitiesPerMinute;
+
         @Override
         public String toString () {
             return String.format(
@@ -508,24 +548,33 @@ public String toString () {
      * and no serious errors, we use a success error code.
      * TODO distinguish between warnings and errors - we already distinguish between info and warnings.
      * This could be turned into a GridJsonBlock constructor, with the JSON writing code in an instance method.
+     * Note that this is very similar to the RegionalWorkResult constructor, and some duplication of logic could be
+     * achieved by somehow merging the two.
      */
     public static void addJsonToGrid (
             OutputStream outputStream,
-            AccessibilityResult accessibilityResult,
+            OneOriginResult oneOriginResult,
             List<TaskError> scenarioApplicationWarnings,
             List<TaskError> scenarioApplicationInfo,
-            PathResultSummary pathResult
+            TransitLayer transitLayer // Only used if oneOriginResult contains paths, can be null otherwise
     ) throws IOException {
         var jsonBlock = new GridJsonBlock();
         jsonBlock.scenarioApplicationInfo = scenarioApplicationInfo;
         jsonBlock.scenarioApplicationWarnings = scenarioApplicationWarnings;
-        if (accessibilityResult != null) {
-            // Due to the application of distance decay functions, we may want to make the shift to non-integer
-            // accessibility values (especially for cases where there are relatively few opportunities across the whole
-            // study area). But we'd need to control the number of decimal places serialized into the JSON.
-            jsonBlock.accessibility = accessibilityResult.getIntValues();
+        if (oneOriginResult != null) {
+            if (oneOriginResult.accessibility != null) {
+                // Due to the application of distance decay functions, we may want to make the shift to non-integer
+                // accessibility values (especially for cases where there are relatively few opportunities across the whole
+                // study area). But we'd need to control the number of decimal places serialized into the JSON.
+                jsonBlock.accessibility = oneOriginResult.accessibility.getIntValues();
+            }
+            if (oneOriginResult.paths != null) {
+                jsonBlock.pathSummaries = new PathResultSummary(oneOriginResult.paths, transitLayer);
+            }
+            if (oneOriginResult.density != null) {
+                jsonBlock.opportunitiesPerMinute = oneOriginResult.density.opportunitiesPerMinute;
+            }
         }
-        jsonBlock.pathSummaries = pathResult;
         LOG.debug("Travel time surface written, appending {}.", jsonBlock);
         // We could do this when setting up the Spark handler, supplying writeValue as the response transformer
         // But then you also have to handle the case where you are returning raw bytes.
@@ -539,10 +588,13 @@ public static void addJsonToGrid (
      * Also returns any accumulated work results to the backend.
      * @return a list of work tasks, or null if there was no work to do, or if no work could be fetched.
      */
-    public List<RegionalTask> getSomeWork () {
+    public List<RegionalTask> getSomeWork (int tasksToRequest) {
+        LOG.debug("Polling backend to report status and request up to {} tasks.", tasksToRequest);
         String url = brokerBaseUrl + "/poll";
         HttpPost httpPost = new HttpPost(url);
         WorkerStatus workerStatus = new WorkerStatus(this);
+        workerStatus.maxTasksRequested = tasksToRequest;
+        workerStatus.pollIntervalSeconds = POLL_INTERVAL_MAX_SECONDS;
         // Include all completed work results when polling the backend.
         // Atomically copy and clear the accumulated work results, while blocking writes from other threads.
         synchronized (workResults) {
@@ -552,7 +604,7 @@ public List<RegionalTask> getSomeWork () {
 
         // Compute throughput in tasks per minute and include it in the worker status report.
         // We poll too frequently to compute throughput just since the last poll operation.
-        // TODO reduce polling frequency (larger queue in worker), compute shorter-term throughput.
+        // We may want to reduce polling frequency (with larger queue in worker) and compute shorter-term throughput.
         workerStatus.tasksPerMinuteByJobId = throughputTracker.getTasksPerMinuteByJobId();
 
         // Report how often we're polling for work, just for monitoring.
@@ -573,8 +625,10 @@ public List<RegionalTask> getSomeWork () {
                 // Broker returned some work. Use the lenient object mapper to decode it in case the broker is a
                 // newer version so sending unrecognizable fields.
                 // ReadValue closes the stream, releasing the HTTP connection.
-                return JsonUtilities.lenientObjectMapper.readValue(responseEntity.getContent(),
-                        new TypeReference<List<RegionalTask>>() {});
+                return JsonUtilities.lenientObjectMapper.readValue(
+                        responseEntity.getContent(),
+                        new TypeReference<List<RegionalTask>>() {}
+                );
             }
             // Non-200 response code or a null entity. Something is weird.
             LOG.error("Unsuccessful polling. HTTP response code: " + response.getStatusLine().getStatusCode());
diff --git a/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorkerTask.java b/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorkerTask.java
index d1e88f78c..f3305e280 100644
--- a/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorkerTask.java
+++ b/src/main/java/com/conveyal/r5/analyst/cluster/AnalysisWorkerTask.java
@@ -99,6 +99,20 @@ public abstract class AnalysisWorkerTask extends ProfileRequest {
      */
     public boolean includePathResults = false;
 
+    /**
+     * Whether to include the number of opportunities reached during each minute of travel in results sent back
+     * to the broker. Requires both an origin and destination pointset to be specified, and in the case of regional
+     * analyses the origins must be non-gridded, and results will be collated to CSV.
+     * It should be possible to enable regional results for gridded origins as well.
+     */
+    public boolean includeTemporalDensity = false;
+
+    /**
+     * If this is set to a value above zero, report the amount of time needed to reach the given number of
+     * opportunities from this origin (known technically as "dual accessibility").
+     */
+    public int dualAccessibilityThreshold = 0;
+
     /** Whether to build a histogram of travel times to each destination, generally used in testing and debugging. */
     public boolean recordTravelTimeHistograms = false;
 
diff --git a/src/main/java/com/conveyal/r5/analyst/cluster/RegionalWorkResult.java b/src/main/java/com/conveyal/r5/analyst/cluster/RegionalWorkResult.java
index 39a650d3e..d15531c03 100644
--- a/src/main/java/com/conveyal/r5/analyst/cluster/RegionalWorkResult.java
+++ b/src/main/java/com/conveyal/r5/analyst/cluster/RegionalWorkResult.java
@@ -37,6 +37,12 @@ public class RegionalWorkResult {
      */
     public int[][][] accessibilityValues;
 
+    /**
+     * The temporal density of opportunities - how many are reached during each minute of travel.
+     * Quantities of opportunities for each [destinationGrid, percentile, minute].
+     */
+    public double[][][] opportunitiesPerMinute;
+
     /**
      * If this field is non-null, the worker is reporting an error that compromises the quality of the result at this
      * origin point, and potentially for then entire regional analysis. Put into a Set on backend since all workers
@@ -59,6 +65,7 @@ public RegionalWorkResult(OneOriginResult result, RegionalTask task) {
         this.travelTimeValues = result.travelTimes == null ? null : result.travelTimes.values;
         this.accessibilityValues = result.accessibility == null ? null : result.accessibility.getIntValues();
         this.pathResult = result.paths == null ? null : result.paths.summarizeIterations(PathResult.Stat.MINIMUM);
+        this.opportunitiesPerMinute = result.density == null ? null : result.density.opportunitiesPerMinute;
         // TODO checkTravelTimeInvariants, checkAccessibilityInvariants to verify that values are monotonically increasing
     }
 
diff --git a/src/main/java/com/conveyal/r5/analyst/cluster/TravelTimeResult.java b/src/main/java/com/conveyal/r5/analyst/cluster/TravelTimeResult.java
index c79f23ab8..c851afcac 100644
--- a/src/main/java/com/conveyal/r5/analyst/cluster/TravelTimeResult.java
+++ b/src/main/java/com/conveyal/r5/analyst/cluster/TravelTimeResult.java
@@ -37,6 +37,9 @@ public class TravelTimeResult {
      * This is optional, generally used only for debugging and testing, so may be null if no histograms are recorded.
      * Since this is used for debugging and testing it might even be interesting to do it at one-second resolution or
      * just store every travel time, but binning makes it easy to compare with probability distributions.
+     * Note that this is one histogram _per target_ showing on how many iterations each travel time is the fastest,
+     * _not_ one histogram per origin/percentile showing how many destinations are reached at each travel time. The
+     * latter is essentially the discrete derivative of step-function accessibility and is tracked elsewhere (TemporalDensityResult).
      */
     int[][] histograms;
 
diff --git a/src/main/java/com/conveyal/r5/analyst/cluster/WorkerStatus.java b/src/main/java/com/conveyal/r5/analyst/cluster/WorkerStatus.java
index 7d002fa9a..1fcd17a6e 100644
--- a/src/main/java/com/conveyal/r5/analyst/cluster/WorkerStatus.java
+++ b/src/main/java/com/conveyal/r5/analyst/cluster/WorkerStatus.java
@@ -24,6 +24,8 @@
 public class WorkerStatus {
 
     private static final Logger LOG = LoggerFactory.getLogger(WorkerStatus.class);
+    private static final int LEGACY_WORKER_MAX_TASKS = 16;
+    public static final int LEGACY_WORKER_MAX_POLL_INTERVAL_SECONDS = 2 * 60;
 
     public String architecture;
     public int processors;
@@ -50,6 +52,20 @@ public class WorkerStatus {
     public String ipAddress;
     public List<RegionalWorkResult> results;
 
+    /**
+     * Then maximum number of tasks the broker should send to this worker. May be zero if its work queue is full.
+     * Default value determines the number of tasks to send to older workers that don't send this value when they poll.
+     */
+    public int maxTasksRequested = LEGACY_WORKER_MAX_TASKS;
+
+    /**
+     * The maximum amount of time the worker will wait before polling again. After this much time passes the backend
+     * may consider the worker lost or shut down. The backend should be somewhat lenient here as there could be delays
+     * due to connection setup, API contention etc. The default value reflects how how long the backend should wait to
+     * hear from older workers that don't send this value.
+     */
+    public int pollIntervalSeconds = LEGACY_WORKER_MAX_POLL_INTERVAL_SECONDS;
+
     /** No-arg constructor used when deserializing. */
     public WorkerStatus() { }
 
diff --git a/src/main/java/com/conveyal/r5/common/Util.java b/src/main/java/com/conveyal/r5/common/Util.java
index ddfce26d5..c79e1c168 100644
--- a/src/main/java/com/conveyal/r5/common/Util.java
+++ b/src/main/java/com/conveyal/r5/common/Util.java
@@ -3,6 +3,7 @@
 import java.util.Arrays;
 import java.util.Collection;
 import java.util.Map;
+import java.util.function.Supplier;
 
 public abstract class Util {
 
@@ -66,4 +67,19 @@ public static int[] newIntArray (int length, int defaultValue) {
         return array;
     }
 
+    /**
+     * Convenience method to create a 2D array and fill it immediately with new instances of a class.
+     * The supplier can be a method reference to a constructor like ToBeInstantiated::new, and the returned
+     * array will be of that type.
+     */
+    public static <T> T[][] newObjectArray (int d1, int d2, Supplier<T> supplier) {
+        T[][] result = (T[][]) new Object[d1][d2];
+        for (int x = 0; x < d1; x++) {
+            for (int y = 0; y < d2; y++) {
+                result[x][y] = supplier.get();
+            }
+        }
+        return result;
+    }
+
 }
diff --git a/src/main/resources/com/conveyal/gtfs/storage/gpwv3-quarter-buffer-boolean.asc b/src/main/resources/com/conveyal/gtfs/storage/gpwv3-quarter-buffer-boolean.asc
new file mode 100644
index 000000000..d1beddacd
--- /dev/null
+++ b/src/main/resources/com/conveyal/gtfs/storage/gpwv3-quarter-buffer-boolean.asc
@@ -0,0 +1,578 @@
+ncols        1440
+nrows        572
+xllcorner    -180.000000000000
+yllcorner    -58.000000000000
+cellsize     0.250000000000
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