COMP9334 Capacity Planning of Computer Systems and Networks
Assignment (Version 1.0), Term 1, 2021

2828秒1728秒2665秒

（a）确定core-1，core-2和磁盘的服务需求。

（b）当有30个交互式用户并且每个作业的思考时间为15秒时，使用瓶颈分析来确定系统吞吐量的渐近界限。

（a）为上述系统建立一个连续时间马尔可夫链，该系统具有3个工作人员和2个等待时间。您的表述应包括状态的定义以及状态之间的转换率。

（b）写下您制定的连续时间马尔可夫链的平衡方程式。

（c）您制定的连续时间马尔可夫链的稳态概率的推导表达式。

（d）确定到达的查询将被拒绝的可能性。

（e）确定队列中已接受查询的平均等待时间。

Question 1 (3 marks)

An interactive computer system consists of a dual-core CPU and a disk. We will use core-1 and core-2 to refer to the two cores of the CPU. The system was monitored for 60 minutes and the following measurements were taken:

Number of completed jobs 1347 Number of accesses to core-1 2087 Number of accesses to core-2 2348 Number of disk accesses 2412

Busy time of core-1 Busy time of core-2 Disk busy time

2828 seconds 1728 seconds 2665 seconds

(a) Determine the service demands of core-1, core-2 and the disk.

(b) Use bottleneck analysis to determine the asymptotic bound on the system throughput when there are 30 interactive users and the think time per job is 15 seconds.

Question 2 (7 marks)

A call centre has 3 staff to deal with customer enquires. The centre has an automatic dis- patcher to direct the calls to the staff. The dispatcher has a queue that can hold up to 2 calls but there are no queueing facilities at the staff’s terminals. The queueing network at the support centre is depicted in Figure 1.

The centre receives on average 12.7 queries per hour. The arrivals can be modelled by using the Poisson distribution.

Each staff can complete on average 4.1 queries per hour. The amount of time required by each query is exponentially distributed.

When a query arrives at the dispatcher, it will accept the query if the dispatcher queue is not full, otherwise the query will be rejected. If a query is accepted and the queue is not empty, the query will be placed at the end of the queue. If a query is accepted and the queue is empty, then the query will be placed in the queue if all staff are busy, otherwise it will be sent to an idling staff. A query will leave the system after its processing is completed. When- ever a staff becomes idle, he/she will take the query from the front of the queue if there is one.

(a) Formulate a continuous-time Markov chain for a system described above with 3 staff and 2 waiting slots. Your formulation should include the definition of the states and the transition rates between states.

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

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

(d)  Determine the probability that an arriving query will be rejected.

(e)  Determine the mean waiting time of an accepted query in the queue.

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