Page 294 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 91 QUEUEING THEORY

Rearranging the above relations

Letting At -, 0, we obtain

9.3. Derive Eqs. (9.13) and (9.1 4).
Setting un = 1, do = 0, and dn = y in Eq. (9.1 O), we get

Pl=-P -
0 - PPo
C1

where po is determined by equating

from which we obtain

9.4. Derive Eq. (9.15).
Since pn is the steady-state probability that the system contains exactly n customers, using Eq. (9.14),
the average number of customers in the M/M/1 queueing system is given by

where p = 1/y < 1. Using the algebraic identity

we obtain

9.5. Derive Eqs. (9.1 6) to (9.1 8).
Since 1, = A, by Eqs. (9.2) and (9.15), we get