الدرس 69 · موضوعات خاصة: نظرية الطوابير

Many servers: M/M/c and Erlang C

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A web server with four cores, a call centre with four agents: many servers share one line. Measure traffic in erlangs, use the Erlang C formula to find the chance that a request has to wait and how long it waits, then see why one shared line beats a separate line for each server.

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01Requests arrive at 600 per second and each takes 5 ms of a server's time on average. How much traffic is that?

A = arrival rate × mean service time = 600 × 0.005 s = 3 erlangs: on average three servers are busy all the time.

02What does the Erlang C formula give you?

Erlang C is P(wait) for an M/M/c queue with unlimited waiting room. The chance of being turned away when there is no waiting room is Erlang B.

03With 4 servers, 320 arrivals a second and a 10 ms mean service time, Erlang C is 0.596. What is the mean wait in the line?

Wq = C ÷ (cμ − λ) = 0.596 ÷ (4 × 100 − 320) per second = 7.46 ms. 12.5 ms is the mean wait of only those requests that do wait.

04Four servers are each 80% busy. Why does one shared line give much shorter waits than four separate lines?

With separate lines a request can wait behind a busy server while another server is idle. Pooling removes that: here the mean wait drops from 40.0 ms to 7.46 ms, about 5.4 times shorter.

05Keep every server 80% busy but grow the pool from 4 to 64 servers. What happens to the chance of waiting?

Bigger pools absorb random bursts better: P(wait) is 59.6% with 4 servers, 30.5% with 16 and 5.6% with 64, all at 80% utilisation.

المصطلحات الأساسية

المسرد

M/M/c: Kendall's notation for a queue with random (Poisson) arrivals, random (exponential) service times and c servers sharing one first-in, first-out line.
Server: Whatever does the work: a CPU core, a worker thread, a link, or an agent in a call centre.
Erlang: The unit of traffic intensity: one erlang keeps one server busy all the time. A = arrival rate × mean service time (ITU-T E.600).
Utilisation (ρ): The share of time each server is busy: ρ = A ÷ c. It must stay below 1, or the line grows without limit.
Erlang C: The probability that an arrival finds all c servers busy and has to wait, for a system with unlimited waiting room.
Erlang B: The probability that an arrival finds all servers busy when there is no waiting room, so it is lost (blocked).
Pooling: Letting several servers share one queue, so no server idles while work waits. At the same load it cuts waiting sharply.
Little's law: Average number waiting = arrival rate × average wait (L = λW). It holds for any stable queue.

تفاصيل إضافية

الملاحظات والمصادر

The 320 requests a second, 10 ms per request and the 64 calls an hour of 3 minutes are illustrative, chosen to give 3.2 erlangs; every other number follows from the M/M/c formulas.
M/M/c assumes random (Poisson) arrivals and exponential service times. Real traffic is often burstier, which makes waits longer, but the pooling effect still holds.
The separate lines take requests at random. Sending each request to the shortest line helps, but it still loses to one shared line, because a request cannot switch to a core that frees up later.
The figure shows a few requests at a time; the meter gives the long-run averages.
Erlang B is the related formula for systems with no waiting room, where a call that finds every line busy is lost.

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