Page 297 - Schaum's Outlines - Probability, Random Variables And Random Processes

QUEUEING THEORY [CHAP. 9

(b) From Eq. (9.1 7),

from which we obtain R 2 0.3 per minute. Thus, the required average arrival rate to justify the second
booth is 18 per hour.

9.10. Derive Eqs. (9.20) and (9.21).
From Eqs. (9.1 9) and (9.1 O), we have

Let p = I/(sp). Then Eqs. (9.46) and (9.47) can be rewritten as

which is Eq. (9.21). From Eq. (9.1 I), p, is obtained by equating

Using the summation formula

we obtain Eq. (9.20); that is,

provided p = A/(sp) < 1.

9.11. Consider an M/M/s queueing system. Find the probability that an arriving customer is forced to
join the queue.
An arriving customer is forced to join the queue when all servers are busy-that is, when the number
of customers in the system is equal to or greater than s. Thus, using Eqs. (9.20) and (9.21), we get
w sS (SPY
P(a customer is forced to join queue) = pn = p0 - 1 pn = po -
n=s S! ,=, s!(l - p)

Equation (9.49) is sometimes referred to as Erlang's delay (or C) formula and denoted by C(s, Alp). Equation
(9.49) is widely used in telephone systems and gives the probability that no trunk (server) is available for an
incoming call (arriving customer) in a system of s trunks.

9.12. Derive Eqs. (9.22) and (9.23).