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Queueing Simulation Models. Example 13.9. Queueing simulation models. A popular alternative to using the analytical models is to develop queueing simulations. There are several advantages to using simulation .
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Queueing Simulation Models Example 13.9
Queueing simulation models • A popular alternative to using the analytical models is to develop queueing simulations. • There are several advantages to using simulation. • Probably the most important advantage is that you are not restricted to the assumptions required by the standard analytical queueing models. • A second advantage of queueing simulation is that you get to see the action through time. • The downside of queueing simulation is that it has traditionally required a clever computer programmer, a specialized software package, or both.
Queueing simulation models continued • The first simulation model we examine is a variation of the M/M/squeueing model from section 13.5. (See the file Multiserver Simulation.xlsm.) • Customers arrive at a service center according to a Poisson process (exponential interarrival times), they wait (if necessary) in a single queue, and then they are served by the first available server.
Queueing simulation models continued • The simulation model is different in the following respects from the analytical M/M/smodel: • The service times are not necessarily exponentially distributed. The file allows three options: • constant (nonrandom) service times, • exponentially distributed service times, and • gamma-distributed service times, as shown below.
Queueing simulation models continued • The waiting room is of limited size, where this size is an input parameter. If the queue is already this long and another customer arrives, this new customer is not allowed to enter the system. • The simulated run time is another user input. • Every time you run the simulation, you are asked for a random number seed. The actual number you enter is not important. The important part is that if you enter the same seed for two different runs, you get the same stream of random numbers.
Queueing simulation models continued • These last two points enable some very important insights into queueing systems in general. • An analytical model such as the M/M/smodel provides summary measures, typically means, in steady state. • But if you simulate such a system for a set amount of time, and average the times in queue for the simulated customers, will the average be the same? The answer is a very definite no. • First, the average might not be the steady-state value because two hours might not be long enough to “get into” steady state. • Second, different runs using different random numbers will typically provide different averages. • This model is shown in the following example.
Example 13.9:Background information • CountyBank has already used analytical models to obtain steady-state measures of queuing behavior. • However, it wonders whether these provide very realistic estimates of what occurs during a 2-hour peak period at the bank. • During the peak period, arrivals occur according to a Poisson process of 2 per minute, there are 6 tellers employed, and each service time has a mean length of 2.7 minutes.
Example 13.9 continued:Background information • The standard deviation of service time is estimated at 1.5 minutes and a histogram of historical service times has a shape much like the shape of the graph shown on the next slide, so that the gamma distribution is reasonable. • What insights can the bank manager obtain from simulation?
Example 13.9 continued:Solution • If we use the M/M/s model, we obtain the results on the next slide. • For example, the mean wait in queue is WQ = 3.33 minutes. • If we use the G/G/s model we obtain the results on the slide after the next. • The value in WQ is now 2.18. • Evidently, the gamma distribution, which has a much lower coefficient of variation, results in less time in the queue.
Example 13.9 continued:Multiserver Simulation.xlsm • The queuing simulation results are shown on the next slide. • This sheet is set up with a nice user interface. • By clicking on the button, you see a couple of dialog boxes where you can enter the required inputs. • They appear on the next slide, with results to follow.
Example 13.9 continued:Solution • Again, we will not delve into all the details but when the simulation runs it does the following: • Starts with an empty and ideal system – no customers in the bank. • Keeps simulating customer arrivals and service times, and it keeps playing out the events, but it doesn’t keep track of any customer statistics for the first 120 minutes, the warmup time. It keeps track of statistics for the next 120 minutes, the run time. • If a customer arrives and there are already 10 customers in line, this customer is turned away. If we want to ensure that no one is turned away, we can choose a large value for this input. • Reports the summary measures for this run as shown.
Example 13.9 continued:Solution • The outputs shown on the next slide should be self-explanatory. • During the 2-hour period, 223 customers entered the bank, and 2 were turned away. • Each teller was busy, on, average, 83.4% of the time, the average customer waited in the queue for 1.06 minutes, the average length was 1.86, the maximum queue length distribution, shown on the next slide. • Each bar represents the percentage of simulated time the queue length was any particular value.
Example 13.9 continued:Solution • Clearly, the average time in queue, 1.06 minutes, is much smaller than WQ from the M/M/s and G/G/s models. • Which is the “correct” value for the County Bank’s 2-hour peak period? • This is not an easy question to answer. • The 1.06 value from the simulation depends to a great extent on the random numbers we happened to simulate.
Example 13.9 continued:Solution • To illustrate this, we ran the simulation several more times, each with a different random number seed, and we obtained values ranging from slightly under 0.7 to slightly over 2.1. • This shows the bank manager that the average time in queue during any day’s 2-hour peak period depends on the day. • Some days she will get lucky and other days she won’t. This variability from day to day – that is, from run to run – is possibly the most important insight we can gain from simulation.
Example 13.9 continued:Solution • Besides the variability from day to day, the simulation results can depend on the run time length, and they can be affected by the limited queue size. • If all customers are allowed to enter the system, the average time in queue increases, whereas if many are turned away, the average time in queue, for those who enter, is much smaller.
