Page 301 - Schaum's Outlines - Probability, Random Variables And Random Processes

QUEUEING THEORY [CHAP. 9

which is Eq. (9.36). From Eq. (9.1 I), p, is obtained by equating
K
n=O
Using the summation formula (9.56), we obtain

which is Eq. (9.35).

9.18. Derive Eq. (9.37).
Using Eq. (9.36) and (9.51), the average number of customers in the queue is given by

n - s)pn-' = po SI
=Po7 C( (sp)" K-s mpm
n=s m=O

9.19. Consider an M/M/s/s queueing system. Find the probability that all servers are busy.
Setting K = s in Eqs. (9.60) and (9.61), we get

and p, is obtained by equating

Thus

The probability that all servers are busy is given by

Note that in an M/M/s/s queueing system, if an arriving customer finds that all servers are busy, the
customer will turn away and is lost. In a telephone system with s trunks, p, is the portion of incoming calls
which will receive a busy signal. Equation (9.64) is often referred to as Erlang's loss (or B) formula and is
commonly denoted as B(s, Alp).

9.20. An air freight terminal has four loading docks on the main concourse. Any aircraft which arrive
when all docks are full are diverted to docks on the back concourse. The average aircraft arrival