[fix](profile) Add parser time to total time of summary profile (#55506)

Author: zhiqiang-hhhh

fe/fe-core/src/main/java/org/apache/doris/common/profile/SummaryProfile.java

@@ -352,6 +352,8 @@ public class SummaryProfile {
 
     private Map<Backend, Long> assignedWeightPerBackend;
 
+    public boolean parsedByConnectionProcess = false;
+
     public SummaryProfile() {
         init();
     }

fe/fe-core/src/main/java/org/apache/doris/qe/ConnectProcessor.java

@@ -347,6 +347,7 @@ public void executeQuery(String originStmt) throws Exception {
                 executor = new StmtExecutor(ctx, parsedStmt);
                 executor.getProfile().getSummaryProfile().setParseSqlStartTime(parseSqlStartTime);
                 executor.getProfile().getSummaryProfile().setParseSqlFinishTime(parseSqlFinishTime);
+                executor.getProfile().getSummaryProfile().parsedByConnectionProcess = true;
                 ctx.setExecutor(executor);
 
                 if (cacheKeyType != null) {

fe/fe-core/src/main/java/org/apache/doris/qe/StmtExecutor.java

@@ -303,7 +303,14 @@ private Map<String, String> getSummaryInfo(boolean isFinished) {
         // reference the implementation of DebugUtil.getPrettyStringMs to figure out the format
         if (isFinished) {
             builder.endTime(TimeUtils.longToTimeString(currentTimestamp));
-            builder.totalTime(DebugUtil.getPrettyStringMs(currentTimestamp - context.getStartTime()));
+            long executionCosts = currentTimestamp - context.getStartTime();
+            // Execution of parser could happen before StmtExecutor is involved.
+            if (getSummaryProfile().parsedByConnectionProcess) {
+                builder.totalTime(DebugUtil.getPrettyStringMs(
+                        executionCosts + getSummaryProfile().getParseSqlTimeMs()));
+            } else {
+                builder.totalTime(DebugUtil.getPrettyStringMs(executionCosts));
+            }
         }
         String taskState = "RUNNING";
         if (isFinished) {
@@ -747,6 +754,7 @@ private void parseByNereids() {
             getProfile().getSummaryProfile().setParseSqlStartTime(System.currentTimeMillis());
             statements = new NereidsParser().parseSQL(originStmt.originStmt, context.getSessionVariable());
             getProfile().getSummaryProfile().setParseSqlFinishTime(System.currentTimeMillis());
+            getProfile().getSummaryProfile().parsedByConnectionProcess = false;
         } catch (Exception e) {
             throw new ParseException("Nereids parse failed. " + e.getMessage());
         }