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470 CHAPTER 11 QUEUING MODELS

service. For example, consider the case of a unloaders who operate the service channel. In
company that owns and operates the trucks used this case, the cost of having the trucks wait and
to deliver goods to and from its manufacturing the cost of operating the service channel are
plant. In addition to the costs associated with the direct expenses to the firm. An economic analysis
trucks waiting to be loaded or unloaded, the firm of the queuing system is highly recommended for
also pays the wages of the truck loaders and these types of situations.

11.6 Other Queuing Models

D.G. Kendall suggested in 1953 a notation that is helpful in classifying the wide
variety of different queuing models that have been developed. The three-symbol
Kendall notation is as follows:

A=B=k

where

A denotes the probability distribution for the arrivals
B denotes the probability distribution for the service time
k denotes the number of channels

Depending on the letter appearing in the A or B position, a variety of waiting
line systems can be described. The letters that are commonly used are as
follows:

M designates a Poisson probability distribution for the arrivals or an exponential
probability distribution for service time
D designates that the arrivals or the service time is deterministic or constant
G designates that the arrivals or the service time has a general probability
distribution with a known mean and variance

Using the Kendall notation, the single-channel waiting line model with Poisson
arrivals and exponential service times is classified as an M/M/1 model. The two-
channel waiting line model with Poisson arrivals and exponential service times
presented in Section 11.3 would be classified as an M/M/2 model.

NOTES AND COMMENTS

n some cases, the Kendall notation is extended to and the fifth symbol is necessary when the popula-
I five symbols. The fourth symbol indicates the larg- tion of arriving units or customers is finite. When the
est number of units that can be in the system, and fourth and fifth symbols of the Kendall notation are
the fifth symbol indicates the size of the population. omitted, the waiting line system is assumed to have
The fourth symbol is used in situations where the infinite capacity, and the population is assumed to
queue can hold a finite or maximum number of units be infinite.

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