【正文】 executives,and a pany shop may perform repairs as needed on the firm’s 20 trucks.Number of servers (channels)The capacity of queuing system is function of the capacity of each server and the number of servers being used. The terms server and channel are synonymous,and it is generally assumed that each channel can handle one customer at a time. Systems can be either singleor multiplechannel.(A group of servers working together as a team,such as a surgical team,is treated as a singlechannel system.) Examples of singlechannel systems are small grocery stores with one checkout counter,some theaters,singlebay car washes, and drivein banks with one teller. Multiplechannel systems(those with more than one server) are monly found in banks, at airline ticket counters,at auto service centers,and at gas stations.A related distinction is the number of steps or phases in a queuing system. For example,at theme parks, people go from one attraction to another. Each attraction constitutes a separate phase where queues can (and usually do) from.Figure 3 illustrates some of the most mon queuing systems. Because it would not be possible to cover all of these cases in sufficient detail in the limited amount of space available here,our discussion will focus on singlephase systems.Single channel,Single phaseSingle channel,multiple phaseMultiple channelsingle phaseMultiple channelmultiple phase Figure 3 Four mon variations of queuing systemsArrival and service patternsWaiting lines are a direct result of arrival and service variability. They occur because random, highly variable arrival and service patterns cause systems to be temporarily overloaded. In many instances,the variabilities can be described by theoretical distributions. In fact, the most monly used models assume that the customer arrival rate can be described by a Possion distribution and that the service time can be describde by a negative exponential distribution. Figure 4 illustrates these distributions.The Poisson distribution often provides a reasonably good description of customer arrivals per unit of time(.,per hour). Figure 5A illustrates how poissondistributed arrivals (., accidents) might occur during a threeday period. In some hours, there are three or four arrivals,in other hours one or two arrivals,and in some hours no arrivals.The negative exponential distribution often provides a reasonably good description of customer service times(.,first aid care for accident victims). Figure 5B illustrates how exponential service times might appear for the customers whose arrivals are illustrated in Figure 5A . Note that most service times are very short—some are close to zero—but a few require a relatively long service time. That is typical of a negative exponential distribution.Waiting lines are most likely to occur when arrivals are bunched or when service times are particularly lengthy, and they are very likely to occur when both factors are present. For instance, note the long service time of customer 7 on day 1, in Figure 5B. In Figure 5A, the seventh customer arrived just after10 o’clock and the next two customers arrived shortly after that, making it very likely that a waiting line formed. A similar situation occurred on day 3 with the last three customers: The relatively long service time for customer 13 (Figure 5B), and the short time before the next two arrivals (Figure 5A,day 3) would create (or increase the length of ) a waiting line.It is interesting to note that the Poisson and negative exponential distributions are alternate ways of presenting the same basic information. That is ,if service time is exponential ,then the service rate is Poisson..Similarly, if the customer arrival rate is Poisson, then the interarrival time (.,the time between arrivals) is exponential. For example, if a service facility can process 12 customers per hour(rate), average service time is five if the arrival rate is 10 per hour, then the average time between arrivals is six minutes.The models described here generally require that arrival and service rates lend themselves to description using a Poisson distribution of ,equivalently, that interarrival and service times lend themselves to description using a negative exponential distribution. In practice, it is necessary to verify that these assumptions are met. Sometimes this is done by collecting data and plotting them, although the preferred approach is to use a chisquare goodnessoffit test for that purpose. A discussion of the chisquare test is beyond the scope of this text, but most basic statistics textbooks cover the topic.Research has shown that these assumptions are often appropriate for customer arrivals but less likely to be appropriate for service. In situations where the assumptions are not reasonably satisfied, the alternatives would be to (1) develop a more suitalbe model, (2) search for a better (and usually more plex) existing model, or (3) resort to puter simulation. Each of these alternatives requires more effort or cost than the ones presented here.Figure 4 Poisson and negative exponential distributionsFigure 5 Poisson arrivals and exponential service timesQueue disciplineQueue discipline refers to the order in which customers are processed. All but one of the models to be described shortly assume that service is provide