Q1. Consider a single-server queuing system for which the inter arrival times are exponentially distributed. A customer who arrives and finds the server busy joins the end of a single queue. Service times of customers at the server are also exponentially distributed random variables. Upon completing service for a customer, the server chooses a customer from 'the queue (if any) in a FIFO manner:
a. Simulate customer arrivals assuming that the mean inter arrival time equals the mean service time (e.g., consider that both of these mean values are equal to 1 min). Create a plot of number of customers in the queue (y-axis) versus simulation time (x-axis). Is the system stable? (Hint: Run the simulation long enough [e.g., 10,000 min] to be able to determine whether or not the process is stable.)
b. Consider now that the mean inter arrival time is 1 min and the mean service time is 0.7 min. Simulate customer arrivals for 5000 min and calculate (i) the average waiting time in the queue, 00 the maximum waiting time in the queue, (H) the maximum queue length, (iv) the proportion of customers having a delay time in excess of 1 min, and (v) the expected utilization of the server.
Q2. Airline ticket counter-At an airline ticket counter, the current practice is to allow queues to form before each ticket agent Time between arrivals to the agents is exponentially distributed with a mean of 5 min. Customers join the shortest queue at the time of their arrival. The service time for the ticket agents is uniformly distributed between 2 and 10 min:
a. Develop an ExtendSim model to determine the minimum number of agents that will result in an average waiting time of 5 min or less.
b. The airline has decided to change the procedure involved in processing customers by the ticket agents. A single line is formed, and customers are routed to the ticket agent who becomes free next. Modify the simulation model in part (a) to distributed between 1 and 4 min with a mode of 3 min. Simulate the operation of the bank for an 8 h period (7 h for the inside tellers). Assess the performance of the current system.
Q3. Grocery store-You are hired by Safeway to help them build a number of simulation models to better understand the customer flows and queuing processes in a grocery store setting. The pilot project at hand focuses on an off-peak setting where at most two checkouts are open. To better understand the necessary level of detail and the complexities involved, Safeway wants a whole battery of increasingly more realistic and complex models. For each model, Safeway wants to keep track of (i.e., plot) the average cycle time, queue length, and waiting time in the queue. To understand the variability, they also want to see the standard deviation of these three metrics. In addition, they would like to track the maximum wait¬ing time and the maximum number of customers in line. Furthermore, to better understand the system dynamics, plots of the actual queue lengths over time are required features of the model. The off-peak setting is valid for about 4 h, so it is reasonable to run the simulation for 240 min. Furthermore, to facilitate an easier first-cut comparison between the models, a fixed random seed is recommended. Because Safeway plans to use these different models later, it is important that each model sheet has a limit of one model:
a. In the first model, your only interest is the queues building up at the checkout counters. Empirical investigation has indicated that it is reasonable to model the arrival process (to the checkout counters) as a Poisson process with a constant arrival intensity of three customers per minute. The service time in a checkout station is on average 30 s per customer and will, in this initial model, be considered constant. Inspired by the successes of a local bank, Safeway wants to model a situation with one single line to both checkout counters. As soon as a checkout is available, the first person in the queue will go to this counter. After the customers have paid for their goods, they immediately leave the store.
b. Upon closer investigation, it is clear that the service time is not constant but rather normally distributed with mean = 30 S and standard deviation =10 s. What is the effect of the additional variability compared to the results in part (a)?
c. To be able to analyze the effect of different queuing configurations, Safeway wants a model in which each checkout counter has its own queue. When a customer arrives to the checkout point, he or she will choose the shortest line. The customer is not allowed to switch queues after making the initial choice. Can you see any differences in system performance compared to the results in part (b)?
d. To make the model more realistic, Safeway also wants to include the time customers spend in the store walking around and picking up their groceries. Empirical investigation has shown that there are basically two types of customers, and they need to be treated somewhat differently.