Monodomain model
In queueing theory, a discipline within the mathematical theory of probability, an M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation.[1] Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory.[2][3] An extension of this model with more than one server is the M/D/c queue.
Model definition
An M/D/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers in the system, including any currently in service.
- Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
- Service times are deterministic time D (serving at rate μ = 1/D).
- A single server serves customers one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
- The buffer is of infinite size, so there is no limit on the number of customers it can contain.
Delay
Define ρ = λ/μ as the utilization; then the mean delay in the system in an M/D/1 queue is[4]
and in the queue:
Busy period
The busy period is the time period measured from the instant a first customer arrives at an empty queue to the time when the queue is again empty. This time period is equal to D times the number of customers served. If ρ < 1, then the number of customers served during a busy period of the queue has a Borel distribution with parameter ρ.[5][6]
Finite capacity
Stationary distribution
A stationary distribution for the number of customers in the queue and mean queue length can be computed using probability generating functions.[7]
Transient solution
The transient solution of an M/D/1 queue of finite capacity N, often written M/D/1/N, was published by Garcia et al in 2002.[8]
References
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