Consider a two$-$server queueing system, where the customers arriving into the system

join a single queue, wait for their turn, receive service from either one of the two

servers and leave the system. The first server is faster than the second one. If an

arriving customer finds both servers available, then it uses the first server. Otherwise,

it simply uses whichever server is available. We are interested in simulating the

behavior of this system with the intention of estimating

the proportion of time that the first server is busy;

the proportion of time both servers are busy; and

the proportion of time that there are one or more customers in the queue over the

first $100$ minutes.

The interarrival times for the customers are exponentially distributed with mean $0\mathrm{.}5$

minutes. The service times at the first server are exponentially distributed with mean

$0\mathrm{.}8$ minutes and the service times at the second server are exponentially distributed

with mean $0\mathrm{.}9$ minutes. We assume that the queue can accommodate infinite number

of customers and the system starts empty.

a$)$ Draw a diagram of this two$-$server queue.

b$)$ Describe the state of the system and the events that change the state of the

system.

c$)$ Write precise pseudo code that describes how different events change the state

of the system and what other events are scheduled in response.

d$)$ Use a worksheet to manually simulate the system over $10$ minutes. For the

interarrival times, use the sequence $1\mathrm{.}3,1\mathrm{.}1,0\mathrm{.}6,1\mathrm{.}5,2\mathrm{.}2,1\mathrm{.}1,0\mathrm{.}5,0\mathrm{.}7,0\mathrm{.}1,1\mathrm{.}4,$

$1\mathrm{.}2,1\mathrm{.}4,1\mathrm{.}4,1\mathrm{.}2,2\mathrm{.}1.$ For the service times at the first server, use the sequence

$2\mathrm{.}1,1\mathrm{.}3,0\mathrm{.}9,1\mathrm{.}4,1\mathrm{.}2,2\mathrm{.}1,2\mathrm{.}2.$ For the service times at the second server, use the

sequence $2\mathrm{.}2,1\mathrm{.}6,1\mathrm{.}1,1\mathrm{.}7,2\mathrm{.}1,1\mathrm{.}3,0\mathrm{.}9.$ If you run out of interarrival and service

times, then make up your own. Make sure that you compute the statistics of

interest in your manual simulation. Use appropriate variables to compute the

statistics $"$on the fly" rather than finishing the whole simulation and looking at

the whole worksheet later. Clearly report the final statistics.

e$)$ Write a computer program that simulates the two$-$server queueing system with

the probability distributions given above. Turn in your $($commented$)$ computer

program. Make sure that your computer program computes the statistics of

interest. Run your computer program $20$ times and use the output from the $20$

runs of the simulation to construct estimates and confidence intervals for the

statistics of interest.