Queuing notation, default values, elements

Queuing (Kendall) notation

A queue is written as A/B/c/K/N/D:

Codes for A and B: M = exponential (Markovian, Poisson arrivals), D = deterministic, E_k = Erlang-k, G = general.

Example 1: M/M/1. Poisson arrivals, exponential service, one server. Measures with arrival rate lambda and service rate mu: utilization rho = lambda/mu, average number in system L = rho/(1 - rho), average time in system W = 1/(mu - lambda). For lambda = 4, mu = 6 per hour: rho = 0.667, L = 2, W = 0.5 hour.

Example 2: M/G/3/30/infinity/FCFS. Poisson arrivals, general service-time distribution, 3 servers, at most 30 customers in the system, infinite population, first come first served.

Default values

The last three fields are often omitted. When omitted, the defaults are:

• K (capacity) = infinity.
• N (population) = infinity.
• D (discipline) = FCFS.

So M/M/2 means M/M/2/infinity/infinity/FCFS, and M/D/1/20 means capacity 20 with infinite population and FCFS. A field can be left out only from the right; the first three fields must always be given.

Elements of a queue system

 Population --> Arrival --> Queue --> Server(s) --> Departure

1. Calling population: the group from which customers come. It may be infinite (public) or finite (machines in a factory).
2. Arrival process: the pattern of arrival, described by the inter-arrival time distribution and its mean rate, single or batch.
3. Queue (waiting line): where customers wait. It may have limited or unlimited capacity.
4. Queue discipline: the rule for selecting the next customer, e.g. FCFS, LIFO, priority or random.
5. Service mechanism: the server or servers, the service-time distribution and the arrangement (parallel or series).
6. Departure: customers leaving after service (or after balking or reneging).
7. Customer behaviour: balking, reneging and jockeying.