Previous Table of Contents Next


The asymptotic bounds are useful in practice since they can be easily derived and explained to people who do not have any background in queueing theory or performance analysis.

The following examples illustrate the use of asymptotic bounds.

Example 33.7 For the timesharing system considered in Example 33.2:
DCPU = 5, DA = 4, DB = 3, Z = 18
D = DCPU + DA + DB = 5 + 4+ 3 = 12
Dmax = DCPU = 5
The asymptotic bounds are

These bounds are shown by broken lines in Figure 33.4. The solid lines show the exact throughput and response times (obtained using the mean-value analysis to be discussed in Section 34.2). The knee occurs at

Thus, if there are more than six users on the system, there will certainly be queueing in the system.
Example 33.8 How many terminals can be supported on the timesharing system of Example 33.2 if the response time has to be kept below 100 seconds?


FIGURE 33.4  Asymptotic bounds on the throughput and the response time.

Using the asymptotic bounds on the response time, we get

R(N) ≥ max{12,5N - 18}

The response time will be more than 100 if

5N - 18 ≥ 100

That is, if

N ≥ 23.6

the response time is bound to be more than 100. Thus, the system cannot support more than 23 users if a response time of less than 100 is required.

This completes the discussion on operational laws. A summary of all operational laws is presented in Box 33.1.

Box 33.1 Operational Laws
Utilization law Ui = XiSi = XDi
Forced flow law Xi = XVi
Little's law Qi = XiRi
General response time law R =
Interactive response time law R = N/X - Z
Asymptotic bounds R ≥ max{D,NDmax - Z}
X ≤ min{1/Dmax,N/(D + Z)}

Symbols:
D Sum of service demands on all devices, ςiDi
Di Total service demand per job for the ith device, = SiVi
Dmax Service demand on the bottleneck device, = maxi{Di}
N Number of jobs in the system
Qi Number in the ith device
R System response time
Ri Response time per visit to the ith device
Si Service time per visit to the ith device
Ui Utilization of the ith device
Vi Number of visits per job to the ith device
X System throughput
Xi Throughput of the ith device
Z Think time

EXERCISES

33.1  During a 10-second observation period, 400 packets were serviced by a gateway whose CPU can service 200 pps. What was the utilization of the gateway CPU?
33.2  The throughput of a timesharing system was observed to be five jobs per second over a 10-minute observation period. If the average number of jobs in the system was four during this period, what was the average response time?
33.3  During a 10-second observation period, 40 requests were serviced by a file server. Each request requires two disk accesses. The average service time at the disk was 30 milliseconds. What was the average disk utilization during this period?
33.4  A distributed system has a print server with a printing speed of 60 pages per minute. The server was observed to print 500 pages over a 10-minute observation period. If each job prints five pages on the average, what was the job completion rate of the system?
33.5  For a timesharing system with two disks (user and system), the probabilities for jobs completing the service at the CPU were found to be 0.80 to disk A, 0.16 to disk B, and 0.04 to the terminals. The user think time was measured to be 5 seconds, the disk service times were 30 and 25 milliseconds, and the average service time per visit to the CPU was 40 milliseconds. Using the queueing network model shown in Figure 32.8, answer the following for this system:
a.  For each job, what are the visit ratios for CPU, disk A, and disk B?
b.  For each device, what is the total service demand?
c.  If disk A utilization is 60%, what is the utilization of the CPU and disk B?
d.  If the utilization of disk B is 10%, what is the average response time when there are 20 users on the system?
33.6  For the system of Exercise 33.5, answer the following:
a.  What is the bottleneck device?
b.  What is the minimum average response time?
c.  What is the maximum possible disk A utilization for this configuration?
d.  What is the maximum possible throughput of this system?
e.  What changes in CPU speed would you recommend to achieve a response time of 10 seconds with 25 users? Would you also need a faster disk A or disk B?
f.  Write the expressions for asymptotic bounds on throughput and response time.
33.7  For the system of Exercise 33.6, which device would be the bottleneck if
a.  the CPU is replaced by another unit that is twice as fast?
b.  disk A is replaced by another unit that is twice as slow?
c.  disk B is replaced by another unit that is twice as slow?
d.  the memory size is reduced so that the jobs make 20 times more visits to disk B due to increased page faults?


Previous Table of Contents Next

Copyright © John Wiley & Sons, Inc.