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456 CHAPTER 11 QUEUING MODELS

Management scientists have found that if the probability distribution for the
service time can be assumed to follow an exponential probability distribution, using
an exponential probability distribution, the probability that the service time will be
less than or equal to a time of length t is given by:

Pðservice time tÞ¼ 1 e t (11:3)

where
¼ the mean number of units that can be served per time period
e ¼ 2:71828

A property of the Suppose that the Dome studied the order-taking and order-filling process and
exponential probability
distribution is that there found that the single food server can process an average of 60 customer orders per
is a 0.6321 probability hour. On a one-minute basis, the mean service rate would be ¼ 60 customers/60
that the random variable minutes ¼ one customer per minute. For example, with ¼ 1, we can use equation
takes on a value less (11.3) to calculate probabilities such as the probability an order can be processed in
than its mean. In queuing
applications, the ½ minute or less, one minute or less and two minutes or less. These calculations are:
exponential probability 1ð0:5Þ
distribution indicates that Pðservice time 0:5 min:Þ¼ 1 e ¼ 1 0:6065 ¼ 0:3935
approximately 63 per Pðservice time 1:0 min:Þ¼ 1 e 1ð1:0Þ ¼ 1 0:3679 ¼ 0:6321
cent of the service times
are less than the mean Pðservice time 2:0 min:Þ¼ 1 e 1ð2:0Þ ¼ 1 0:1353 ¼ 0:8647
service time and
approximately 37 per So we would conclude that there is a 0.3935 probability that an order can be
cent of the service times processed in ½ minute or less, a 0.6321 probability that it can be processed in one
are greater than the minute or less and a 0.8647 probability that it can be processed in two minutes or less.
mean service time.
In several waiting line models presented in this chapter, we assume that the
probability distribution for the service time follows an exponential probability dis-
tribution. In practice, you should collect data on actual service times to determine
whether the exponential probability distribution is a reasonable approximation of
the service times for your application.

Steady-State Operation
When the Dome restaurant opens in the morning, no customers are in the restau-
rant. Gradually, activity builds up to a normal or steady state. The beginning or start-
up period is referred to as the transient period. The transient period ends when the
system reaches the normal or steady-state operation. Waiting line models describe
the steady-state operating characteristics of a waiting line.

Single-Channel Queuing Model with Poisson Arrivals
11.2
and Exponential Service Times
Queuing models are often
based on assumptions In this section we present formulas that can be used to determine the steady-state
such as Poisson arrivals
and exponential service operating characteristics for a single-channel queuing system. The formulas are
times. When applying any applicable if the arrivals follow a Poisson probability distribution and the service
waiting line model, data times follow an exponential probability distribution. As these assumptions apply to
should be collected on the the Dome problem introduced in Section 11.1, we show how formulas can be used to
actual system to ensure
that the assumptions of determine the Dome’s operating characteristics and thus provide management with
the model are reasonable. helpful decision-making information.

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