Queuing system and Kendall's notation

Queuing system

A queuing (delay) system in telecommunication is one in which a call or packet that finds all servers (trunks, registers, processors, links) busy is not lost but waits in a queue until a server becomes free. Examples: calls waiting for an operator in a call centre, digits waiting for a common-control register, packets waiting in a router buffer. Its performance is measured by probability of delay, mean waiting time and queue length, rather than by lost-call probability.

 arrivals    +----------+     +-----------+   departures
 ---lambda-->|  queue   |---->| N servers |--------->
             | (buffer) |     |  (mu each)|
             +----------+     +-----------+

Characteristics in Kendall's notation

Kendall's notation describes a simple queue as A / B / C / K / N / D (often shortened to A/B/C):

1. Arrival pattern: usually Poisson with mean rate $\lambda$ calls per unit time, so inter-arrival times are exponential.
2. Service pattern: usually exponential holding time with mean $h = 1/\mu$.
3. Number of servers working in parallel.
4. System capacity: limited buffers make the system partly a loss system.
5. Source population: large (infinite) or small (finite) number of subscribers.
6. Discipline: order in which waiting calls are served.

Examples

• M/M/1: Poisson arrivals, exponential service, one server — e.g. a single public phone; utilisation $\rho = \lambda/\mu$, mean number waiting $L_q = \rho^2/(1-\rho)$.
• M/M/N (Erlang C): N trunks with queuing, used for call centres.
• M/M/N/N: no waiting room — the Erlang B loss system.
• M/D/1: fixed packet length on a link.