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).