Consider a single-server queuing system for which the interarrival times are exponentially distributed. A customer who arrives a

Consider a single-server queuing system for which the interarrival 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 interarrival 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 interarrival 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.

8.3 A service facility consists of two servers in series (tandem), each with its own FIFO

queue (see Figure 8.63). A customer completing service at server 1 proceeds to server 2,

and a customer completing service at server 2 leaves the facility. Assume that the interarrival

times of customers to server 1 are exponentially distributed with mean of 1 min.

Service times of customers at server 1 are exponentially distributed with a mean of

0.7 min, and at server 2 they are exponentially distributed with a mean of 0.9 min:

a. Run the simulation for 1000 min and estimate for each server the expected average

waiting time in the queue for a customer and the expected utilization.

b. Suppose that there is a travel time from the exit of server 1 to the arrival to queue 2

(or server 2). Assume that this travel time is distributed uniformly between 0 and

2 min. Modify the simulation model and run it again to obtain the same performance

measures as in part (a). (Hint: You can add an Activity block to simulate this

travel time. A uniform distribution can be used to simulate the time.)

c. Suppose that no queuing is allowed for server 2. That is, if a customer completing

service at server 1 sees that server 2 is idle, the customer proceeds directly

to server 2, as before. However, a customer completing service at server 1 when

server 2 is busy with another customer must stay at server 1 until server 2 gets

done; this is called blocking. When a customer is blocked from entering server 2,

the customer receives no additional service from server 1 but prevents server 1

from taking the first customer, if any, from queue 1. Furthermore, new customers

may arrive to queue 1 during a period of blocking. Modify the simulation model

and rerun it to obtain the same performance measures as in part (a).

8.12 Assessing process performance—The process of insuring a property consists of four

main activities: review and distribution, underwriting, rating, and policy writing.

Four clerks, three underwriting teams, eight raters, and five writers perform these

activities in sequence. The time to perform each activity is exponentially distributed

with an average of 40, 30, 70, and 55 min, respectively. On the average, a total of

40 requests per day are received. Interarrival times are exponentially distributed.

A flowchart of the process is depicted in Figure 8.67:

a. Develop a simulation model of this process. The model should simulate 10 days

of operation. Assume that work in process (WIP) at the end of each day becomes

the beginning WIP for the next day.

b. Add data collection to calculate the following measures: resource utilization,

waiting time, length of the queues, WIP at the end of each day, and average daily

throughput (given in requests per day).

c. Assess the performance of the process with the data collected in part (b).