10. Solving Queueing Models

10. Solving Queueing Models - CS4 Modelling and Simulation...

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CS4 Modelling and Simulation LN-10 10 Solving Queueing Models 10.1 Introduction In this note we look at the solution of systems of queues, starting with simple isolated queues. The beneFts of using predeFned, easily classiFed queues will become appar- ent: many performance measures can be calculated directly from the parameters of the model. Obviously the situation becomes more complicated when queues are connected to- gether. However we see that even in this case deriving performance measures can be very straightforward as we are able to consider each queue in isolation. ±inally we summarise the assumptions which we need to make in order to obtain these solutions. 10.2 Single Queues Since we assume that all the queues which we consider have a Markovian arrival process and a Markovian service process the queue can be modelled as a Markov process. The state transition diagram for a single-server queue with inFnite capacity is shown in ±igure 22. Note that unlike the Markov processes which we have considered earlier in the course this process has an inFnite state space. x 0 x 1 x 3 x 2 ... µ µµµ λλ λ λ ±igure 22: The state transition diagram for a simple M/M/ 1 queue If we write the global balance equations for this system we can soon recognise a regular pattern emerging. λ π 0 = µ π 1 ( λ + µ ) π 1 = λ π 0 + µ π 2 ( λ + µ ) π 2 = λ π 1 + µ π 3 . . . Using simple algebra we can rewrite these as shown below: π 1 = λ µ π 0 π 2 = λ µ π 1 = ± λ µ ² 2 π 0 π 3 = λ µ π 2 = ± λ µ ² 3 π 0 . . . 68
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CS4 Modelling and Simulation LN-10 Remembering that traffic intensity , ρ is defned to be λ/µ (For the single server case, λ/ ( c × µ ) in the general case) we can see that For an arbitrary state x i π = ρ i π 0 and using the normalisation condition we see that 1= X i =0 π = π 0 X i =0 ρ i = π 0 1 1 ρ iF ρ< 1 ±rom this we can deduce that π 0 =1 ρ and π i =( 1 ρ ) ρ i For all i> 0. Thus we have a symbolic evaluation oF the steady state distribution For any M/M/ 1 queue, i.e. the steady state distribution expressed in terms oF the parameters oF the model, λ and µ . This means that For any particular model we can fnd the steady state distribution simply by evaluating the expressions above with our particular value oF ρ = λ/µ .M o r e importantly, we can derive symbolic expressions For the important perFormance measures we are likely to want to derive From a queue, and then evaluate those measures For a particular model without having to carry out any numerical solution of global balance equations . Utilisation, U The queue is being utilised whenever it is non-empty; in other words the utilisation, U ,is1 π 0 . Utilisation U = ρ Mean number of customers in the queue, N This is the expectation oF the number oF customers in the service Facility as a whole, i.e.
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This note was uploaded on 04/08/2010 for the course COMPUTER E 409232 taught by Professor Mohammadabdolahiazgomiph.d during the Spring '10 term at Islamic University.

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10. Solving Queueing Models - CS4 Modelling and Simulation...

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