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SINGLE-CHANNEL QUEUING MODEL WITH POISSON ARRIVALS AND ARBITRARY SERVICE TIMES 471

Single-Channel Queuing Model with Poisson Arrivals
11.7
and Arbitrary Service Times

Let us return to the single-channel model where arrivals are described by a Poisson
When providing input to probability distribution. However, we now assume that the probability distribution
the M/G/1 model, be for the service times is not an exponential probability distribution. Thus, using the
consistent in terms of the Kendall notation, the queuing model that is appropriate is an M/G/1 model, where G
time period. For example,
if l and are expressed denotes a general or unspecified probability distribution.
in terms of the number of
units per hour, the Operating Characteristics for the M/G/1 Model
standard deviation of the
service time should be The notation used to describe the operating characteristics for the M/G/1 model is:
expressed in hours. The
example that follows ¼ the mean arrival rate
uses minutes as the time
period for the arrival and ¼ the mean service rate
service data. ¼ the standard deviation of the service time
Some of the steady-state operating characteristics of the M/G/1 waiting line model
are as follows:
1 The probability that no units are in the system:

P 0 ¼ 1 (11:24)

2 The average number of units in the waiting line:

2 2
þð = Þ 2
L q ¼ (11:25)
2ð1 = Þ

3 The average number of units in the system:

L ¼ L q þ (11:26)

4 The average time a unit spends in the queue:

L q
W q ¼ (11:27)

5 The average time a unit spends in the system:

1
W ¼ W q þ (11:28)

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