THE APPLICATION OF QUEUEING THEORY:DECISION MAKING

DECISION MAKING

Queueing-type situations that require decision making arise in a wide variety of contexts. For this reason, it is not possible to present a meaningful decision-making procedure that is applicable to all these situations. Instead, this section attempts to give a broad concep- tual picture of a typical approach.

Designing a queueing system typically involves making one or a combination of the following decisions:

1. Number of servers at a service facility.

2. Efficiency of the servers.

3. Number of service facilities.

When such problems are formulated in terms of a queueing model, the corresponding de- cision variables usually are s (number of servers at each facility), (mean service rate per busy server), and (mean arrival rate at each facility). The number of service facilities is directly related to because, assuming a uniform workload among the facilities, equals the total mean arrival rate to all facilities divided by the number of facilities. (Section 17.10 also mentions two other possible decisions when designing a queueing system, namely, the amount of waiting space in the queue and any priorities for different categories of cus- tomers, but we will focus in this chapter on the three types of decisions listed above.)

Refer to Sec. 26.1 and note how the three examples there respectively illustrate situ- ations involving these three decisions. In particular, the decision facing Simulation, Inc., in Example 1 is how many repairers (servers) to provide. The problem for Emerald University in Example 2 is how fast a computer (server) is needed. The problem facing Mechanical Company in Example 3 is how many tool cribs (service facilities) to install as well as how many clerks (servers) to provide at each facility.

The first kind of decision is particularly common in practice. However, the other two also arise frequently, particularly for the internal service systems described in Sec. 17.3. One example illustrating a decision on the efficiency of the servers is the selection of the type of materials-handling equipment (the servers) to purchase to transport certain kinds of loads (the customers). Another such example is the determination of the size of a maintenance crew (where the entire crew is one server). Other decisions concern the number of service facilities, such as copy centers, computer facilities, tool cribs, storage areas, and so on, to distribute throughout an area.

All the specific decisions discussed here involve the general question of the appropriate level of service to provide in a queueing system. As mentioned at the beginning of Chap. 17 and in Sec. 17.10, decisions regarding the amount of service capacity to provide usually are based primarily on two considerations: (1) the cost incurred by providing the service, as

shown in Fig. 26.1, and (2) the amount of waiting time for that service, as suggested in Fig. 26.2. Figure 26.2 can be obtained by using the appropriate waiting-time equation from queueing theory. (For better conceptualization, we have drawn these figures and the subsequent two figures as smooth curves even though the level of service may be a discrete variable.)

These two considerations create conflicting pressures on the decision maker. The ob- jective of reducing service costs recommends a minimal level of service. On the other hand, long waiting times are undesirable, which recommends a high level of service. There- fore, it is necessary to strive for some type of compromise. To assist in finding this com- promise, Figs. 26.1 and 26.2 may be combined, as shown in Fig. 26.3. The problem is thereby reduced to selecting the point on the curve of Fig. 26.3 that gives the best balance between the average delay in being serviced and the cost of providing that service. Reference to Figs. 26.1 and 26.2 indicates the corresponding level of service.

Cost of service per arrival

Obtaining the proper balance between delays and service costs requires answers to such questions as, How much expenditure on service is equivalent (in its detrimental im- pact) to a customer’s being delayed 1 unit of time? Thus, to compare service costs and waiting times, it is necessary to adopt (explicitly or implicitly) a common measure of their impact. The natural choice for this common measure is cost, which then requires estima- tion of the cost of waiting.

Because of the diversity of waiting-line situations, no single process for estimating the cost of waiting is generally applicable. However, we shall discuss the basic considerations involved for several types of situations.

One broad category is where the customers are external to the organization provid- ing the service; i.e., they are outsiders bringing their business to the organization. Con- sider first the case of profit-making organizations (typified by the commercial service systems described in Sec. 17.3). From the viewpoint of the decision maker, the cost of waiting probably consists primarily of the lost profit from lost business. This loss of business may occur immediately (because the customer grows impatient and leaves) or in the future (be- cause the customer is sufficiently irritated that he or she does not come again). This kind of cost is quite difficult to estimate, and it may be necessary to revert to other criteria, such as a tolerable probability distribution of waiting times. When the customer is not a human being, but a job being performed on order, there may be more readily identifiable costs incurred, such as those caused by idle in-process inventories or increased expediting and administrative effort.

Now consider the type of situation where service is provided on a nonprofit basis to customers external to the organization (typical of social service systems and some transportation service systems described in Sec. 17.3). In this case, the cost of waiting usually is a social cost of some kind. Thus, it is necessary to evaluate the consequences of the waiting for the individuals involved and/or for society as a whole and to try to impute a monetary value to avoiding these consequences. Once again, this kind of cost is quite dif- ficult to estimate, and it may be necessary to revert to other criteria.

A situation may be more amenable to estimating waiting costs if the customers are internal to the organization providing the service (as for the internal service systems dis- cussed in Sec. 17.3). For example, the customers may be machines (as in Example 1) or employees (as in Example 3) of a firm. Therefore, it may be possible to identify directly some of or all the costs associated with the idleness of these customers. Typically, what is being wasted by this idleness is productive output, in which case the waiting cost be- comes the lost profit from all lost productivity.

Given that the cost of waiting has been evaluated explicitly, the remainder of the analysis is conceptually straightforward. The objective is to determine the level of service that minimizes the total of the expected cost of service and the expected cost of waiting for that ser- vice. This concept is depicted in Fig. 26.4, where WC denotes waiting cost, SC denotes ser- vice cost, and TC denotes total cost. Thus, the mathematical statement of the objective is to

Minimize E(TC) = E(SC) + E(WC).

The next three sections are concerned with the application of this concept to various types of problems. Thus, Sec. 26.3 describes how E(WC) can be expressed mathemati- cally. Section 26.4 then focuses on E(SC) to formulate the overall objective function E(TC) for several basic design problems (including some with multiple decision variables, so that the level-of-service axis in Fig. 26.4 then requires more than one dimension). This section also introduces the fact that when a decision on the number of service facilities is required, time spent in traveling to and from a facility should be included in the analysis (as part of the total time waiting for service). Section 26.5 discusses how to determine the expected value of this travel time.