Question

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, (ii) the maximum waiting time in the queue, (iii) 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.

Answer 1

To simulate customer arrivals in a single-server queuing system with exponentially distributed interarrival times and service times, a Monte Carlo simulation approach can be used. For part A, the simulation can be run for a long duration to determine the system's stability. For part B, several performance measures can be calculated based on specific parameters.

Step-by-step explanation:

To simulate customer arrivals in a single-server queuing system with exponentially distributed interarrival times and service times, we can use a Monte Carlo simulation approach. In this simulation, we generate random numbers from the exponential distribution to represent the interarrival and service times. We start with an empty queue and a server that is initially idle. The simulation progresses by generating random interarrival times and service times, and updating the queue and server accordingly.

For part A of the question, we need to simulate customer arrivals assuming that the mean interarrival time equals the mean service time. We can set the mean interarrival and service times to 1 minute and run the simulation for a long enough duration (e.g., 10,000 minutes) to determine whether the process is stable. We can plot the number of customers in the queue against simulation time to visualize the system's stability.

For part B, we need to consider a different set of parameters: a mean interarrival time of 1 minute and a mean service time of 0.7 minutes. We simulate customer arrivals for 5000 minutes and calculate several performance measures: average waiting time in the queue, maximum waiting time in the queue, maximum queue length, proportion of customers with a delay time exceeding 1 minute, and the expected utilization of the server.