Assignment 4

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Assignment 4

Question 1 [4+ 4+ 4 + 4 + 4 =20 points]

Briefly describe in your Word answer file how you can consider and analyze each of the following scenarios as a queueing system. In particular, for each  scenario  (a) to (e) determine the corresponding calling population, the customers or items that enter the system for service, the servers, the nature of the service, the capacity of the queue, and the queue discipline.

(a) A hair salon

(b) A parking lot

(c) A fire station

(d) A group of machines assigned to an operator for maintenance

(e) A supercomputer shared among a group of researchers at a university.

Question 2 [10 + 20 + 10 = 40 points]

Consider a B&D queuing system which has a finite number of states denoted by n = 0, 1, 2, 3, 4. The arrival rates are λn= 4 − nfor n = 0, 1, 2, 3. The service rates are μn= nfor n = 1, 2, 3, 4. Both rates are expressed using the sametime unit such as arrivals and services per week. Answer to a) to c) in your Word file.

(a) Give an example of a queuing system which can be described by the arrival and service rates given above.

(b) Calculate the steady-state probabilities {Pn } of the system states using the general approach discussed in class, as well as in the textbook. Determine the expected queue length, the expected arrival rate, and the expected waiting time.

(c) Consider this queuing system in its steady-state condition. Assume that if the system is in state n, then the operating cost per day is f(n) = n2  (costs are expressed in thousands of dollars). Determine the average operating cost per day.

Question 3 [20 + 20 = 40 points]

Consider the Excel file General L1-L2-s Simulator, discussed in class and posted on Canvas. Apply the following modeling assumptions:

The number of servers is set to 5.

The interarrival times follow the exponential distribution, with mean value 10 per hour.

The service times follow a uniform distribution, with minimum value of 5 minutes, and a set maximum value parameter STMV (Service Time Maximum Value), expressed in minutes. In separate simulation runs, set the parameter STMV starting from 5, in 5-unit steps to 50 (i.e., set SMTV sequentially to 5, 10, … , 50).

(a) For each considered value of STMV, set the number of simulated customer arrivals to 10000. Record the resulting estimates of the expected total queue length (ETQL) and the expected waiting time (EWT).

(b) Based on your results, create two line plots to express the relationship between ETQL and SMTV, and between EWT and SMTV. Propose a suitable regression model. (You can create these plots and models by using the charting features of Excel.) Briefly summarize your findings in your Word answer file.