本次计算机网络代写的主要内容是网络调度马尔科夫链相关
COMP9334 Capacity Planning of Computer Systems and Networks
Assignment (Version 1.0), Term 1, 2021
问题1（3分）
交互式计算机系统由双核CPU和磁盘组成。我们将使用core-1和core-2来指代CPU的两个内核。监视系统60分钟，并进行以下测量：
完成的作业数1347对core-1的访问数2087对core-2的访问数2348磁盘访问数2412
核心1的繁忙时间核心2的繁忙时间磁盘繁忙时间
回答以下问题。
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。
注意：如果您使用计算机程序来导出数字答案，则必须在提交中包括计算机程序。不要忘了向我们展示您获得答案的步骤。
问题2（7分）
呼叫中心有3名员工来处理客户询问。该中心有一个自动分配器，可将呼叫定向到工作人员。调度员的队列最多可容纳2个呼叫，但工作人员终端上没有排队设施。支持中心的排队网络如图1所示。
该中心每小时平均接收12.7个查询。可以使用泊松分布对到达进行建模。
每个员工平均每小时可以完成4.1个查询。每个查询所需的时间呈指数分布。
当查询到达调度程序时，如果调度程序队列未满，它将接受查询，否则查询将被拒绝。如果查询被接受并且队列不为空，则查询将放置在队列的末尾。如果查询被接受并且队列为空，那么如果所有工作人员都忙，则查询将被放入队列中，否则将其发送给空闲的工作人员。处理完成后，查询将离开系统。每当有工作人员闲置时，他/她都会从队列的最前面进行查询（如果有的话）。
回答以下问题：
（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
Answer the following questions.
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.
Note: If you use a computer program to derive your numerical answers, you must include your computer program in your submission. Do not forget to show us your steps to obtain your answer.
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.
Answer the following questions:
(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.