Queuing Theory

Summary of results

2

Notations

• Typical performance characteristics of queuing models are:

L : Ave. number of customers in the system

LQ : Ave. number of customers waiting in queue

W : Ave. time customer spends in the system

WQ: Ave. time customer spends waiting in the queue

3

Queue notation

M/M/k/c

Arrival processM = MarkovianGI = General

Departure process(Service time distribution)M = MarkovianG = General

Number of servers

Capacity of the queueIf nothing is specified,we assume infinite capacity

4

M/M/1 queue

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5

M/M/1/N queue

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6

M/M/1/N queue

• Expected amount of time spent by the customer…. But, what is a customer?

• Do we include those who came but didn’t join the queue because it was full?

• Or, are we including only those who actually joined the system?

• Depending on our consideration, λa = λ in the first case; whereas λa=λ(1-PN) in the second case. Then,

a

LW

7

Tandem queues

1 2M/M/1 M/M/1

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8

M/G/1

ondistributiy probabilit a having timeservice Random :

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Derivation of these results is slightly more tedious because, unlikeprevious models where the Markov theory was extensively used,M/G/1 model requires Renewal Reward theory.

9

M/G/1: Priority queues

• Let there be two types of customers (Type 1 and 2). Type 1 being the priority class.

• Meaning, service can never begin for Type 2 customer, if Type 1 is waiting in queue. However, preemption is not allowed.

• Let arrival rate of Type 1 customers be λ1 and that of Type 2 be λ2 (both arrivals are Poisson processes).

• Respective service time distribution be G1 and G2. That is S1~ G1 and S2~ G2.

• Once again, we derive the performance characteristics based on the renewal reward theory.

10

M/G/1: Priority queues

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11

M/M/k queues

• Two types of queues could be considered:

1. Loss function queues: Customer does not join the system if (s/)he sees all k servers busy. This queue is called Erlang’s Loss system.

2. Infinite capacity queues. These are exact extensions of M/M/1 queues.

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:parameter system Loss

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12

M/M/k queues

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