M/D/1 queue: Difference between revisions

Revision as of 20:36, 16 May 2016

Revision as of 21:36, 23 May 2016

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==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 customersentities 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 customersentities one at a time from the front of the queue, according to a [[first-come, first-served]] discipline. When the service is complete the customerentity leaves the queue and the number of customersentities in the system reduces by one.

* The buffer is of infinite size, so there is no limit on the number of customersentities it can contain.

The [[state space]] diagram for M/D/1 queue is as below:

[[File:1 Queue.png|none|thumb|637x637px|Stage Space Diagram of M/D/1 Queue]]

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=== Example ===

Customers arriveConsidering a Starbuckssystem linethat athas aonly rateone of 20 per hourserver, and followswith an exponentialarrival distribution.rate Thereof is20 onlyentities oneper server,hour and the service rate is at a constant of 30 per hour.

So the utilization of the server is: ρ=20/30=2/3. Using the metrics shown above, the results are as following: 1) Average number in line L_Q= 0.6667; 2) Average number in system L =1.333; 3) Average time in line ω_Q = 0.033 hour; 4) Average time in system ω = 0.067 hour.

Arrival Rate: 20 per hour

Service Rate: 30 per hour

ρ=20/30=2/3

Using the queueing theory equations, the results are as following:

Average number in line= 0.6667

Average number in system: 1.333

Average time in line: 0.033 hour

Average time in system: 0.067 hour

=== Relation for Mean Waiting Time in M/M/1 and M/D/1 queues ===