
Part 2. MM4XL Tools > 2. Analytical Tools > Business Formulas > 8. Queuing Theory > Queuing Theory Business FormulasQueuing TheoryQueuing theory applies to business problems dealing with waiting lines, such as in banks, post offices, restaurants, theme parks, cinemas, airports, hotels, theatres, and at traffic lights just to mention a few. Waiting lines do not always contain people. Email messages may wait in a line before being routed to their final destination; product parts may wait to be assembled; hamburgers in a fast food restaurant may wait in a line to be sold.Someone or something waiting in a line may carry economic implications that are worth analyzing for business purposes. It can influence the level of customer satisfaction, determining whether customers will decide to repurchase our products and services; or it can increase the production cost structure, thereby making products more or less appealing to buyers. In many cases, managers have a certain level of control over waiting lines. For example, increasing the number of cash registers in a supermarket would reduce the waiting time of customers. Such decisions, however, carry costs, and queuing theory is applied in order to find a tradeoff between the cost of production and the cost of having dissatisfied customers. The concept is summarized in the following chart. Investing in the service level lowers the cost of dissatisfaction, however, beyond a certain optimal level it is not worth investing further. Queuing theory helps find the balance. Queuing System Configurations Waiting lines are configured in many different ways. The following images show the most common ones. The first configuration shows a singlequeue, singleserver system. The customers arrive and join the line waiting to be served in a FIFO order. This configuration is typical of many restaurants, GP offices, cinemas, and so on. The second configuration shows a singlequeue, multiserver system. Here customers wait until they are served by the first available server, whose number may vary depending on the number of customers waiting. This configuration is typical of many post offices, hospitals, grocery stores, airport checkin counters, banks, etc. The last configuration is a collection of singlequeue, singleserver systems and it represents a special case of the first system discussed above. This system is typical of fast food restaurants like Burger King and McDonald's, of cash counters in supermarkets, ticket offices at rail stations, and others. Queuing System Characteristics Two key elements define analytical queuing system models: the kind of customer arrival and the amount of time it takes for them to be served. Arrival Rate In most queuing systems customers arrive at random time intervals. It is appropriate to model such systems with a Poisson distribution function, where the average number of arrivals is denoted with . If, for example, a hypothetical company's hotline receives on average four calls every hour, the following chart shows the probability of receiving any number of calls between zero and 16 during a given hour. The interarrival time, which is the amount of time occurring between calls, follows an exponential probability distribution with average equal to 1/ . Therefore, the hotline of our example can expect, on average, to receive a call every 15 minutes. The exponential distribution plays a key role in queuing theory because it is memoryless (has no memory) and it fits the memoryless property of random arrivals in queue systems, that is, the next arrival is independent from the previous one. It is important to verify that both assumptions of randomness and independence of arrivals hold before modeling a queue system. To do so one could gather some data about the process and plot them on a chart. If the shape of the distributions resembles the shape in the above chart it may be reasonably assumed that the assumptions hold. Otherwise, one can employ goodness of fit tests to verify the assumptions. Service Rate The queue time is the amount of time a customer spends in line waiting to be serviced. The service time is the time spent to receive the service. Service time is often modeled as an exponential function with denoting the average number of customers (or jobs) that can be serviced in a time unit, for instance one hour. Because the exponential is a continuous distribution, the probability of any value is zero. Therefore, probabilities must be associated with time intervals, for example, the probability that the service time will be between t1 and t2. The above chart shows that the probability of occurrence of short service time is higher than the probability of long service time. In this case, the zero service time should be read as a number between more than zero and below a given value. In the literature it has been demonstrated that on average the exponential distribution provides reliable time estimates for modeling queuing systems. In this case too, it is suggested to verify the assumption as discussed for the arrival time. Kendall Notation The Kendall notation was developed to identify the characteristics of the broad variety of existing queuing systems. Simple systems can be denoted using 3 characteristic elements: Nature of the arrival process M = Markovian interarrival time (with an exponential shape) D = Deterministic interarrival time (not random) Nature of service time M = Markovian service time (with an exponential shape) G = General service time (with a nonexponential shape) D = Deterministic service time (not random) The third characteristic is the number of servers in the system. Therefore, an MM1 queue indicates a queue system with exponential interarrival time, exponential service time, and one server available. 
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