# 11.8 Concluding Remarks

### 11.8 Concluding Remarks

Single queue systems provide useful information when estimating waiting times due to contention for shared resources. The M/G/1 queue is one of the most studied queuing systems. A large number of results are available, including vacationing servers and different priority scheduling schemes. Approximate results for G/G/1 and G/G/c are also given. The following chapters consider networks of queues, where individual queues are interconnected .

### 11.9 Exercises

1. Show that in a G/G/1 queue, the average number of customers at the server is equal to the utilization of the server.

2. Derive Eqs. (11.3.9) and (11.3.11) using the Generalized Birth-Death theorem.

3. Derive the average waiting time for M/M/1 from Eq. (11.7.30).

4. Consider two Web clusters, A and B. Cluster A has n servers and cluster B has m (m > n) servers. Requests arrive at each cluster at a rate of l requests/sec. A load balancer in front of each cluster evenly distributes the requests to each server in the cluster. The average service time of a request in cluster A is x seconds and the average service time of a request in cluster B is k x x where k > 1. The service time of a request in either cluster has an arbitrary distribution. Derive an expression for the value of m so that the average response of a request in cluster A is the same as in cluster B.

5. A computer system receives requests from a Poisson process at a rate of 10 requests/sec. Assume that 30% of the requests are of type a and the remaining are of type b . The average service times and the coefficients of variation of the service times for these two types of requests are: E [ S a ] = 0.1 seconds, , E [ S b ] = 0.08 seconds, and . Compute the average response time for each type of request under each of the following scenarios: 1) requests of type a and b have equal priorities, 2) requests of type a have non-preemptive priority over requests of type b , 3) requests of type b have non-preemptive priority over requests of type a , 4) requests of type a have preemptive priority over requests of type b , and 5) requests of type b have preemptive priority over requests of type a .

6. Consider the class 3 requests in Example 11.7 when the server uses a preemptive resume scheduling policy (see Section 11.6.2). It is stated the performance (i.e., the waiting time) of the highest priority requests (i.e., class 3 in this case) is not affected by the lower priority requests. Prove this statement by computing the waiting time of class 3 requests using vanilla M/G/1 results (i.e., from Section 11.4). Compare the result to the value computed in Section 11.6.2.

### Bibliography

[1] D. Gross and C. M. Harris, Fundamentals of Queueing Theory , 3rd ed., Wiley-Interscience, 1998.

[2] L. Kleinrock, Queueing Systems, Vol I: Theory , John Wiley & Sons, New York, 1975.

[3] L. Kleinrock, Queueing Systems, Vol II: Computer Applications , John Wiley & Sons, New York, 1976.

[4] R. Nelson, Probability, Stochastic Processes, and Queuing Theory: The Mathematics of Computer Performance Modelling , Springer Verlag, New York, 1995.

## Chapter 12. Single Class MVA

Section 12.1.  Introduction

Section 12.2.  MVA Development

Section 12.3.  The MVA Algorithm

Section 12.4.  Balanced Systems

Section 12.5.  MVA Extensions and Limitations

Section 12.6.  Chapter Summary

Section 12.7.  Exercises

Bibliography