Grinstead-Snell-Probability · Problem 22

Markov Chains

Mathematics · queueing-service-arrival-distribution

Problem

In the queueing problem of Exercise 20, let

S

be the total service time required by a customer and

T

the time between arrivals of the customers. (a) Show that

P(S = j) = (1-r)^{j-1}r

and

P(T = j) = (1-p)^{j-1}p

, for

j > 0

. (b) Show that

E(S) = 1/r

and

E(T) = 1/p

. (c) Interpret the conditions

s < 1

s = 1

and

s > 1

in terms of these expected values.