feat: add Prometheus metrics instrumentation

- New whisper_live/metrics.py with Counter, Gauge, Histogram metrics
- Track connections (opened/closed/rejected), transcription latency,
  audio processed, segments emitted, REST requests, and errors
- All metric helpers are no-ops when prometheus_client not installed
- --metrics_port CLI flag to expose /metrics endpoint (0 = disabled)
- Metrics integrated into server.py, base.py at key instrumentation points
- 17 new tests in tests/test_metrics.py (178 total passing)

whisper_live/metrics.py

@@ -0,0 +1,122 @@
+"""
+Prometheus metrics for WhisperLive server.
+
+Exposes a /metrics HTTP endpoint on a configurable port for Prometheus scraping.
+All metrics are optional — the server works fine without prometheus_client installed.
+"""
+
+import logging
+import threading
+
+try:
+    from prometheus_client import (
+        Counter,
+        Gauge,
+        Histogram,
+        start_http_server,
+    )
+
+    CONNECTIONS_TOTAL = Counter(
+        "whisperlive_connections_total",
+        "Total WebSocket connections accepted",
+    )
+    CONNECTIONS_ACTIVE = Gauge(
+        "whisperlive_connections_active",
+        "Currently active WebSocket connections",
+    )
+    CONNECTIONS_REJECTED = Counter(
+        "whisperlive_connections_rejected_total",
+        "Connections rejected (server full or auth failure)",
+        ["reason"],
+    )
+    TRANSCRIPTION_LATENCY = Histogram(
+        "whisperlive_transcription_latency_seconds",
+        "Time to transcribe a single audio chunk",
+        buckets=(0.05, 0.1, 0.25, 0.5, 1.0, 2.5, 5.0, 10.0),
+    )
+    AUDIO_PROCESSED = Counter(
+        "whisperlive_audio_processed_seconds_total",
+        "Total seconds of audio processed",
+    )
+    SEGMENTS_EMITTED = Counter(
+        "whisperlive_segments_emitted_total",
+        "Total transcription segments sent to clients",
+        ["completed"],
+    )
+    REST_REQUESTS = Counter(
+        "whisperlive_rest_requests_total",
+        "Total REST API requests",
+        ["endpoint", "status"],
+    )
+    ERRORS = Counter(
+        "whisperlive_errors_total",
+        "Total errors by type",
+        ["type"],
+    )
+
+    _AVAILABLE = True
+
+except ImportError:
+    _AVAILABLE = False
+
+
+def is_available():
+    """Check if prometheus_client is installed."""
+    return _AVAILABLE
+
+
+def start_metrics_server(port=9091):
+    """Start the Prometheus metrics HTTP server on the given port.
+
+    Args:
+        port (int): Port to serve /metrics on. Default 9091.
+    """
+    if not _AVAILABLE:
+        logging.warning("prometheus_client not installed; metrics endpoint disabled")
+        return
+    try:
+        start_http_server(port)
+        logging.info(f"Prometheus metrics available at http://0.0.0.0:{port}/metrics")
+    except Exception as e:
+        logging.error(f"Failed to start metrics server: {e}")
+
+
+def track_connection_opened():
+    if _AVAILABLE:
+        CONNECTIONS_TOTAL.inc()
+        CONNECTIONS_ACTIVE.inc()
+
+
+def track_connection_closed():
+    if _AVAILABLE:
+        CONNECTIONS_ACTIVE.dec()
+
+
+def track_connection_rejected(reason="full"):
+    if _AVAILABLE:
+        CONNECTIONS_REJECTED.labels(reason=reason).inc()
+
+
+def track_transcription_latency(seconds):
+    if _AVAILABLE:
+        TRANSCRIPTION_LATENCY.observe(seconds)
+
+
+def track_audio_processed(seconds):
+    if _AVAILABLE:
+        AUDIO_PROCESSED.inc(seconds)
+
+
+def track_segment_emitted(completed=True):
+    if _AVAILABLE:
+        SEGMENTS_EMITTED.labels(completed=str(completed).lower()).inc()
+
+
+def track_rest_request(endpoint="/v1/audio/transcriptions", status="200"):
+    if _AVAILABLE:
+        REST_REQUESTS.labels(endpoint=endpoint, status=str(status)).inc()
+
+
+def track_error(error_type="transcription"):
+    if _AVAILABLE:
+        ERRORS.labels(type=error_type).inc()