Question 1 -

An interactive computer system consists of three devices: a CPU and three disks (denoted by Disk1, Disk2 and Disk3). The system was monitored for 30 minutes and the following measurements were taken:

Number of completed jobs - 1,231

Number of CPU accesses 2,789

Number of Disk1 accesses - 17,412

Number of Disk2 accesses - 13,424

Number of Disk3 accesses - 15,978

CPU busy time - 1102 seconds

Disk1 busy time - 929 seconds

Disk2 busy time - 1017 seconds

Disk3 busy time - 1265 seconds

Think time - 27 seconds

(a) Determine the service demand at each device of the system.

(b) Use bottleneck analysis to determine the asymptotic bound on the system throughput when there are 40 active terminals.

(c) Using your results in Part (b), compute the minimum possible response time when the number of terminals is 40.

Question 2 -

A call centre has 4 operators. Calls arrive at the call centre obey the Poisson distribution with a rate of 20 calls per hour. The service time required by each call is exponentially distributed with mean service time 10 minutes.

(a) Assuming the call centre has no facilities to place an incoming call on hold. This means that if all the operators are busy, an incoming call will be rejected. Compute the probability that an incoming call is rejected.

(b) The owner of the call centre would like to decrease the call rejection rate to less than 50% of the value calculated in Part (a). The owner decides to achieve this by introducing a queue which places the incoming calls on hold when all the operators are busy.

The holding queue will consist of M holding slots where M is to be determined. If an incoming call arrives when all operators are busy, it will be placed in the holding queue provided that a vacant holding slot is available. If the call arrives when all operators are busy and all M holding slots are used, then the call is rejected. Assuming that the customers are infinitely patient in the sense that once their call is accepted in the (holding) queue, they will wait until they get to the operator and will only leave the system after they have been served.

(i) Formulate a continuous-time Markov chain for the call centre with 4 operators and M holding slots. Your formulation should include the definition of the states and the transition rates between states.

(ii) Write down the balance equations for the continuous-time Markov chain that you have formulated in Part (b, i).

(iii) Derive expressions for the steady state probabilities of the continuous-time Markov chain that you have formulated.

(iv) Use your answer in Part (b,iii) to determine the smallest value of M required to reduce the call rejection rate to less than 50% of the value calculated in Part (a).

(v) For the value of M that you have calculated in Part (b,iv), determine how long an accepted call will need to wait before it will be served by an operator.

Question 3 -

Consider the computer system shown in Figure 1. The system consists of three devices: a disk and 2 CPUs. Each device is modelled as a server and a queue. The system is at peak load and there are three (3) jobs circulating in the system at all times. During each round that a job circulates the system, the job requires processing from one of the CPUs and then followed by the disk. Assuming that:

• The processing time required by each job per visit to the disk is exponentially distributed with mean 50 milli-seconds.
• The two CPUs have different mean processing times. The mean processing times for CPU1 and CPU2 are, respectively, 50 and 100 milli-seconds. Both processing time distributions are assumed to be exponential.
• After a job has left the disk, it will proceed to receive processing at one of the CPUs immediately. In any attempt to utilise the faster CPU (i.e. CPU1), a job will only be sent to CPU2 if it is idle CPU2 is idle and CPU1 is busy. In other words, if CPU2 is busy, the job will be sent to CPU1; and if both CPU1 and CPU2 are idle, the job will be sent to CPU1.

Answer the following questions.

(a) Let the states be the following 3-tuple: (number of users in the CPU1, number of users in CPU2, number of users in the disk), formulate a continuous-time Markov chain for this computer system. Your formulation should include (1) a list of states; (2) the transition rates between the states.

(b) Write down the balance equations for the continuous-time Markov chain that you have formulated in Part (a).

(c) What is the steady state probability for each state?

(d) What is the throughput of the system?

(e) What is the mean response time of the CPU1?

(f) How long does a user have to wait, on average, at the disk before it gets served?