Theories Of Birth-Death Process In Queueing Theory

Birth Death Process in Queueing Theory

Ashwani Kumar

Department of Industrial Engineering and Operation research, IIT Bombay

September 20, 2014

1

INTRODUCTION

A flow of customer from infinite/finite population towards the service facil- ity from a queue(waiting line) on account of lack of capability to serve them all at a time. The queues may be of persons waiting at a doctor’s clinic or at railway ticket booking counter, etc. Bt the mean customer we mean the arriving unit that requires some service to be performed. The customer may be persons, machines, vehicles,parts, etc. Queues stands for a number of customers waiting to be served. The queue does not include the customer being serviced.

The mechanism of queueing process is very simple. Customers arrive at a service counter and are attended by one or more servers. As soon as a customer is served, it departs from the system.

The general framework of a queueing system is shown in Fig. 1.1:

2

Generalized Model of Birth-Death Process

The theory of Birth-Death process, developed by Feller,comprises part of the subject matter commonly called stochastic processes. We now give the model

1of the Birth-Death process. This model deals with a queueing system having

Therefore

Expected rate of inflow at state n = Expected rate of outflow at state n

i.e. λ n−1 P n−1 + μ n+1P n+1 = λ n P n + μ n P n

;for n ≥ 1 and μ 1 P 1 = λ 0 P 0

;for n = 0

Using iterative procedure we have,

P 1 = λ 0 P 0 μ 1

P 2 = (λ 1 +μ 1 )P 1 μ 2

P 3 = λ 2 λ 1 λ 0 P 0 μ 3 μ 2 μ 1

-

λ 0 P 0 μ 2

=

λ 1 λ 0 P 0 μ 1 μ 2

Therefore in general, we can write the following formula:

P n = λ n−1 λ n−2 ......λ 0 P 0

,

μ n μ n−1 ......μ 1

P n = P 0 n−1 λ i i=0 μ i+1 , for n ≥ 1 for n ≥ 1

2Now,

P n+1 =

(λ n +μ n )P n μ n+1

-

λ n−1 P n−1 μ n+1 n λ i i=0 μ i+1 .

= P 0

Thus by mathematical induction the general value of P n holds for all n.

Therefore to obtain the value of P 0 , we use the boundary condition

= 1 or P 0 + ∞ n=1 = 1, to get

P 0 = (1 +

3

∞ n=1 ∞ n=0 P n n−1 λ i −1 i=1 μ i+1 )

Special Cases

Case 1. When λ n = λ for n ≥ 0; and μ n = μ for n > 1

P 0 = [1 +

∞

n −1 n=1 (λ/μ) ]

This result is exactly same as that of M/M/1 model, as in M/M/1 model λ/μ = ρ

Therefore, P 0 = 1 - ρ

And, P n = ρ n (1 − ρ),

Case 2. When λ n =

Therefore P 0 = [1 +

P n = ρ n exp(−ρ)

,

n! for n ≥ 0. λ n+1 for n ≥ 0 and μ n = μ for n > 1.

∞

λ n −1 n=1 n!μ n ] for n ≥ 0 and ρ =

= [1 + ρ + ρ 2

2!

+

ρ