Expose Prometheus metrics on a separate port

Metrics are served on METRICS_PORT (default 2112, "off" to disable), never on
the public port. Besides the Go runtime and process metrics this adds:

- http_requests_total and http_request_duration_seconds per route pattern,
  from an Echo middleware.
- musicserver_songs_requested_total and musicserver_songs_played_total by
  pick mode (random, low, classic). The mode is now stored with each queued
  song (migration 000006, not released yet).
- Library size, sync duration, last successful sync, broken songs from the
  last full sync, and the database connection pool.

Refs #35, #36, #37

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

internal/metrics/pool.go

@@ -0,0 +1,70 @@
+package metrics
+
+import (
+	"github.com/jackc/pgx/v5/pgxpool"
+	"github.com/prometheus/client_golang/prometheus"
+)
+
+// poolCollector reports the database connection pool's statistics on every scrape.
+type poolCollector struct {
+	pool *pgxpool.Pool
+
+	acquiredConns        *prometheus.Desc
+	idleConns            *prometheus.Desc
+	totalConns           *prometheus.Desc
+	maxConns             *prometheus.Desc
+	acquires             *prometheus.Desc
+	acquireWait          *prometheus.Desc
+	emptyAcquires        *prometheus.Desc
+	canceledAcquires     *prometheus.Desc
+	newConns             *prometheus.Desc
+	maxLifetimeDestroyed *prometheus.Desc
+	maxIdleDestroyed     *prometheus.Desc
+}
+
+// NewPoolCollector returns a collector for the musicserver_db_pool_* metrics of pool.
+// Register it once, for example with prometheus.MustRegister.
+func NewPoolCollector(pool *pgxpool.Pool) prometheus.Collector {
+	desc := func(name, help string) *prometheus.Desc {
+		return prometheus.NewDesc("musicserver_db_pool_"+name, help, nil, nil)
+	}
+	return &poolCollector{
+		pool:                 pool,
+		acquiredConns:        desc("acquired_conns", "Connections in use."),
+		idleConns:            desc("idle_conns", "Idle connections."),
+		totalConns:           desc("total_conns", "Open connections, in use, idle and being opened."),
+		maxConns:             desc("max_conns", "Most connections the pool opens."),
+		acquires:             desc("acquires_total", "Connections handed out."),
+		acquireWait:          desc("acquire_wait_seconds_total", "Total time spent waiting for a connection."),
+		emptyAcquires:        desc("empty_acquires_total", "Acquires that had to wait because no connection was idle."),
+		canceledAcquires:     desc("canceled_acquires_total", "Acquires canceled while waiting, for example by a timeout."),
+		newConns:             desc("new_conns_total", "Connections opened."),
+		maxLifetimeDestroyed: desc("max_lifetime_destroyed_total", "Connections closed for reaching their maximum lifetime."),
+		maxIdleDestroyed:     desc("max_idle_destroyed_total", "Connections closed for being idle too long."),
+	}
+}
+
+func (c *poolCollector) Describe(ch chan<- *prometheus.Desc) {
+	prometheus.DescribeByCollect(c, ch)
+}
+
+func (c *poolCollector) Collect(ch chan<- prometheus.Metric) {
+	s := c.pool.Stat()
+	gauge := func(d *prometheus.Desc, v float64) {
+		ch <- prometheus.MustNewConstMetric(d, prometheus.GaugeValue, v)
+	}
+	counter := func(d *prometheus.Desc, v float64) {
+		ch <- prometheus.MustNewConstMetric(d, prometheus.CounterValue, v)
+	}
+	gauge(c.acquiredConns, float64(s.AcquiredConns()))
+	gauge(c.idleConns, float64(s.IdleConns()))
+	gauge(c.totalConns, float64(s.TotalConns()))
+	gauge(c.maxConns, float64(s.MaxConns()))
+	counter(c.acquires, float64(s.AcquireCount()))
+	counter(c.acquireWait, s.AcquireDuration().Seconds())
+	counter(c.emptyAcquires, float64(s.EmptyAcquireCount()))
+	counter(c.canceledAcquires, float64(s.CanceledAcquireCount()))
+	counter(c.newConns, float64(s.NewConnsCount()))
+	counter(c.maxLifetimeDestroyed, float64(s.MaxLifetimeDestroyCount()))
+	counter(c.maxIdleDestroyed, float64(s.MaxIdleDestroyCount()))
+}