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Queuing model; M/D/2/60/150/FIFO meaning

Short answer

A queuing model mathematically describes customers arriving, waiting, and being served, for predicting wait time and length. Figure: calling population -> queue -> server(s) -> departures. M/D/2/60/150/FIFO: Poisson arrivals, deterministic service, 2 servers, capacity 60, finite population 150, first-in first-out. Kendall notation A/B/c/K/N/Z: A inter-arrival dist, B service dist, c servers, K capacity, N population, Z discipline. Example: M/M/1 (exponential inter-arrival/service, 1 server).

What is a queuing model? Explain with figure. What is the meaning of M/D/2/60/150/FIFO in queuing notation? Explain the Kendall notation with example.

Queuing model

A queuing model is a mathematical and logical description of a system in which customers (jobs, packets, patients) arrive, wait in a line if the server is busy, get service and leave. It is used to predict measures such as average waiting time, queue length and server utilisation.

Main elements of the model:

  • Calling population: the source of customers (finite or infinite).
  • Arrival process: how customers arrive, described by the inter-arrival time distribution (for example Poisson arrivals, mean rate lambda).
  • Queue: the waiting area, with a capacity limit and a discipline (FIFO, LIFO, priority).
  • Service mechanism: one or more servers with a service time distribution (mean rate mu).
  • Departure: customers leave after service.

Figure

 Arrivals      +-----------------+     +--------+   Departures
 (calling  --> |  Queue (waiting | --> | Server | -->
 population)   |  line, buffer)  |     | (s)    |
               +-----------------+     +--------+
                  queue discipline      service
                  (FIFO, LIFO...)       time dist.

In a simulation, the clock advances from event to event (arrival, start of service, departure). The state is the number in the system and the server status.

Meaning of M/D/2/60/150/FIFO

The fields are read in the order Arrival / Service / Servers / System capacity / Population / Discipline.

FieldValueMeaning
MMarkovianPoisson arrivals, i.e. exponential inter-arrival times
DDeterministicConstant (fixed) service time
2ServersTwo parallel servers
60System capacityAt most 60 customers in the system (queue plus those in service); others are turned away
150Calling populationFinite source of 150 customers
FIFODisciplineFirst in, first out

Example: 150 registered users send print jobs. Jobs arrive randomly, two printers each take a fixed time per job, at most 60 jobs can be held, and jobs are printed in order of arrival.

Kendall notation

Kendall's notation describes a queue in a short code A/B/c/K/N/Z (Kendall originally used A/B/c; the other three fields were added later). If the last three fields are omitted, they are taken as infinite capacity, infinite population and FIFO.

SymbolPositionMeaningCommon codes
A1stInter-arrival time distributionM (exponential), D (deterministic), Ek (Erlang), G (general)
B2ndService time distributionM, D, Ek, G
c3rdNumber of parallel servers1, 2, 3, ...
K4thSystem capacity (queue + service)finite number or infinity
N5thSize of calling populationfinite number or infinity
Z6thQueue disciplineFIFO, LIFO, SIRO, PRI

Example: M/M/1 means Poisson arrivals, exponential service, one server, infinite capacity and population, FIFO. A single-teller bank counter is a typical M/M/1 queue. M/M/3/20 is a three-server queue where at most 20 customers can be in the system. M/G/1 has general service times, as in a CPU with arbitrary job lengths.

✓ Answer checked by Claude Opus 5.5 (high effort)

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