Page 493 -

MULTIPLE-CHANNEL MODEL WITH POISSON ARRIVALS, ARBITRARY SERVICE TIMES AND NO QUEUE 473

NOTES AND COMMENTS

henever the operating characteristics of a queu- variation in service times results in a larger average
W ing system are unacceptable, managers often number of units in the queue. Hence, another alterna-
try to improve service by increasing the mean service tive for improving the service capabilities of a queue is
rate . This approach is good, but Equation (11.25) to reduce the variation in the service times. Thus, even
shows that the variation in the service times also when the mean service rate of the service facility can-
affects the operating characteristics of the queue. not be increased, a reduction in will reduce the
Because the standard deviation of service times, , average number of units in the queue and improve
appears in the numerator of Equation (11.25), a larger the other operating characteristics of the system.

Multiple-Channel Model with Poisson Arrivals, Arbitrary
11.8
Service Times and No Queue

An interesting variation of the queuing models discussed so far involves a system in
which no waiting is allowed. Arriving units or customers seek service from one of
several service channels. If all channels are busy, arriving units are denied access to
the system. In queuing terminology, arrivals occurring when the system is full are
blocked and are cleared from the system. Such customers may be lost or may
attempt a return to the system later.
A primary application of this model involves the design of telephone and other
communication systems where the arrivals are the calls and the channels are the
number of telephone or communication lines available. In such a system, the calls
are made to one telephone number, with each call automatically switched to an open
channel if possible. When all channels are busy, additional calls receive a busy signal
and are denied access to the system.
The specific model considered in this section is based on the following assumptions.
1 The system has k channels.
2 The arrivals follow a Poisson probability distribution, with mean arrival rate l.
3 The service times for each channel may have any probability distribution.
4 The mean service rate is the same for each channel.
5 An arrival enters the system only if at least one channel is available. An arrival
occurring when all channels are busy is blocked – that is, denied service and
not allowed to enter the system.
With G denoting a general or unspecified probability distribution for service times,
the appropriate model for this situation is referred to as an M/G/k model with
‘blocked customers cleared’. The question addressed in this type of situation is,
How many channels or servers should be used?

Operating Characteristics for the M/G/k Model with Blocked
Customers Cleared
We approach the problem of selecting the best number of channels by calculating
the steady-state probabilities that j of the k channels will be busy. These proba-
bilities are:

Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.