# Limitations of Analytical Models

Analytical queuing models are being applied, advantageously, to many practical situations in operations management. However, when the real life situations are portrayed by an analytical queuing theory model various assumptions creep in. For instance, in our models presented here in we have assumed a convenient Poisson distribution. Analytical models are available for constant service rate. The assumptions regarding any statistical distribution should satisfy two criteria: (a) they should represent the reality fairly well, and (b) they should be such that analytical solutions can be worked out without too much complexity. In many real-life situations, both these criteria cannot be met simultaneously. Can arrivals at a shopping center be approximated by a Poisson distribution?

Many different behavioral patterns have to be considered where arrivals depend upon the state of the queuing system. In addition to finding a suitable statistical distribution, there is the problem of various possible Queue Disciplines. Further the problem of transient (time dependent) solutions as against the simpler steady state solutions exists.

Multiple channels, multiple service problems are even more complex to be handled analytically. This situation is encountered all the time in a make-to-order production business here the jobs have to go through a number of processes. The jobs â€˜waitâ€™ at every work center. The typical problem is one of finding a suitable â€˜dispatch ruleâ€™, which is the â€˜â€™queue disciplineâ€ so as to achieve the desired criteria such as (a) minimum total flow time, or (b) machinery/labor utilization , or (c) minimum standard deviation of the distribution of completion, or (d) minimize the percentage of orders completed late etc.

Analytical solutions are available for two/ three work cases, and that too for a limited be of situations. This is where Digital Simulation by means of a computer facility comes in handy like Job-shop evaluating the stimulated systemâ€™s performance on the desired criteria.

Monte Carlo is an approach for simulating the probability distribution by associating and then selecting random numbers.

In this technique, we can incorporate with more ease many of the real-life constraints. For instance, if the customers go to the shop mostly during the evening or on holidays, such constraints could be incorporated into the Digital Simulation model. Various complex rules encountered in the queuing discipline can also be easily incorporated by means of this technique. We shall explain the basic elements of these techniques by means of an example given below. The same basic principles could be extended to complex problem such as multiple channels and multiple services with a variety of queuing disciplines.

Let us consider over there a shop with one sever who performs one type of service of one customer at a time. Suppose he has observed the following inter-arrival times and their frequencies (that means, the number of times this particular inter-arrival time was observed). Table below not only shows the inter-arrival times and corresponding frequencies but also calculates the probability of occurrences, the cumulative probability, and based on the cumulative probability the associated random numbers.

Now, refer to random number table and pick out, say, 10 random numbers which will be our representative sample. For instance, if our first random number happens to be 81, then the corresponding inter-arrival time is 5 minutes. Similarly, if our second random number happens to be 54, then the second inter-arrival time is 3 minutes (for the example given above), and so on. This way we are creating a simulation sample for the customer arrival rates. Now, depending upon the sample of these arrival rates or the inter-arrival times of the customers of our example we can calculate by simple arithmetic, the number of minutes a customer/number of customer who have to wait in order to be serviced, the amount of time that the server remains idle in our sample, and the number of people on an average waiting to be served in our sample. In this way, we can find the average waiting line and the idle time for the server from our sample data.

Monte Carlo Simulation, as discussed above is useful for complex queuing situations because complex queuing disciplines, complex distributions of inter-arrival times and services times can all be included in the simulation model without much difficulty. Of course, if such a simulation exceeds a certain number of observations, it has to make use of a computer facility. The data on the inter-arrival times and the service times can be stored and a computer program can specify the random selection of the sample as well as the queuing discipline and other important features of the flow problem.