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454 CHAPTER 11 QUEUING MODELS

Figure 11.1 The Dome Single-Channel Waiting Line

System

Server
Customer
Arrivals
Queue Order Taking Customer
and Order Leaves
Filling After Order
Is Filled

Single-Channel Queue
In the current Dome operation, a server takes a customer’s order, determines the
total cost of the order, takes the money from the customer and then fills the order.
Once the first customer’s order is filled, the server takes the order of the next
customer waiting for service. This operation is an example of a single-channel
queuing system. Each customer entering the Dome restaurant must pass through
the one channel – one order-taking and order-filling station – to place an order, pay
the bill and receive the food. When more customers arrive than can be served
immediately, they form a queue and wait for the order-taking and order-filling
station to become available. A diagram of the Dome single-channel system is shown
in Figure 11.1.

Distribution of Arrivals
Defining the arrival process for a waiting line involves determining the probability
distribution for the number of arrivals in a given period of time. For many queuing
situations, the arrivals occur randomly and independently of other arrivals, and we
cannot predict exactly when an arrival will occur. In such cases, quantitative analysts
have found that the Poisson probability distribution provides a good description of
the arrival pattern.
The Poisson probability function provides the probability of x arrivals in a specific
time period. The probability function is as follows. 1

x
e
PðxÞ¼ for x ¼ 0; 1; 2; ... (11:1)
x!
where

x ¼ the number of arrivals in the time period
¼ the mean number of arrivals per time period (pronounced ’lambda’)
e ¼ 2:71828
Values of e l can be found using a spreadsheet, a calculator or by using Appendix B.

1
The term x!, x factorial, is defined as x! ¼ x(x 1)(x 2) . . . (2)(1). For example 4! ¼ (4)(3)(2)(1) ¼ 24. For
the special case of x ¼ 0, 0! ¼ 1 by definition.

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