M/D/1 queue: Difference between revisions

Revision as of 03:22, 24 May 2016

Revision as of 06:11, 24 May 2016

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=== Transition Matrix ===

The transition probability matrix for a M/D/1 queue with arrival rate λ and service time 1, such that λ <1 (for stability of the queue) is given by P as below :<ref name=":1" />:

<math>P=\begin{pmatrix} a_0 & a_1 & a_2 & a_3 & ... \\ a_0 & a_1 & a_2 & a_3 & ...\\ 0 & a_0 & a_1 & a_2 & ...\\ 0&0 & a_0 & a_1 & ...\\... & ... &...&... &...\\\end{pmatrix}</math> ， <math>a_n=\frac{\lambda^n}{n!}e^{-\lambda}</math>, n = 0,1,....

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=== Example ===

Considering a system that has only one server, with an arrival rate of 20 entities per 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.

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

For an equilibrium M/G/1 queue, the expected value of the time W spent by a customer in the queue are given by Pollaczek-Khintchine formula as below:<ref name=":0">{{Cite book|title=Introduction to Queuing Theory|last=Cooper|first=Robert B.|publisher=Elsevier Science Publishing Co.|year=1981|isbn=0-444-00379-7|___location=|pages=189}}</ref>:

<math>E(W)=\frac{\rho\tau}{2(1-\rho)}(1+\frac{\sigma^2}{\tau^2})</math>

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where τ is the mean service time; σ² is the variance of service time; and ρ=λτ < 1, λ being the arrival rate of the customers.

For M/M/1 queue, the service times are exponentially distributed, then σ² = τ² and the mean waiting time in the queue denoted by W_M is given by the following equation:<ref name=":0" />:

<math>{W_M}=\frac{\rho\tau}{1-\rho}</math>

Using this, the corresponding equation for M/D/1 queue can be derived, assuming constant service times. Then the variance of service time becomes zero, i.e. σ² = 0. The mean waiting time in the M/D/1 queue denoted as W_D is given by the following equation:<ref name=":0" />:

<math>{W_D}=\frac{\rho\tau}{2(1-\rho)}</math>