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Substituting for Qi from Equation (33.5) yields

XR = X1R1 + X2R2 + ... + XMRM

Dividing both sides of this equation by X and using the forced flow law, we get

R = V1R1 + V2R2 + ... + VMRM

or

This is called the general response time law. It is possible to show that this law holds even if the job flow is not balanced. Intuitively, the law states that the total time spent by a job at a server is the product of time per visit and the number of visits to the server; and the total time in the system is equal to the sum of total times at various servers.

Example 33.5 Let us compute the response time for the timesharing system of Examples 33.2 and 33.4. For this system

VCPU = 181, VA = 80, VB = 100

RCPU = 0.250, RA = 0.203, RB = 0.071

The system response time is
R = RCPUVCPU + RAVA + RBVB
= 0.250 × 181 + 0.203 × 80 + 0.071 × 100 = 68.6
The system response time is 68.6 seconds.

33.5 INTERACTIVE RESPONSE TIME LAW

In an interactive system, the users generate requests that are serviced by the central subsystem and responses come back to the terminal. After a think time Z, the users submit the next request. If the system response time is R, the total cycle time of requests is R + Z. Each user generates about T/(R + Z) requests in time period T. If there are N users,

This is the interactive response time law.

Example 33.6 For the timesharing system of Example 33.2, we can compute the response time using the interactive response time law as follows:

X = 0.1963, N = 17, Z = 18

Therefore,

This is the same as that obtained earlier in Example 33.5.

33.6 BOTTLENECK ANALYSIS

One consequence of the forced flow law is that the device utilizations are proportional to their respective total service demands:

Ui Di

The device with the highest total service demand Di has the highest utilization1 and is called the bottleneck device. This device is the key limiting factor in achieving higher throughput. Improving this device will provide the highest payoff in terms of system throughput. Improving other devices will have little effect on the system performance. Therefore, identifying the bottleneck device should be the first step in any performance improvement project.


1Delay centers can have utilizations more than one without any stability problems. Therefore, delay centers cannot be a bottleneck device. Only queueing centers should be considered in finding the bottleneck or computing Dmax.

Suppose we find that the device b is the bottleneck. This implies that Db = Dmax is the highest among D1, D2,...,DM. Then the throughput and response times of the system are bound as follows:

Here, is the sum of total service demands on all devices except terminals. Equations (33.8) and (33.9) are known as asymptotic bounds. The proof follows.

Proof 33.1 The asymptotic bounds are based on the following observations:
1.  The utilization of any device cannot exceed 1. This puts a limit on the maximum obtainable throughput.
2.  The response time of the system with N users cannot be less than a system with just one user. This puts a limit on the minimum response time.
3.  The interactive response time formula can be used to convert the bound on throughput to that on response time and vice versa.
We now derive the bounds using these observations. For the bottleneck device b we have

Ub = XDmax

Since Ub cannot be more than 1, we have

With just one job in the system, there is no queueing and the system response time is simply the sum of the service demands:

R(1) = D1 + D2 + ... + DM = D

Here, D is defined as the sum of all service demands. With more than one user there may be some queueing and so the response time will be higher. That is,
R(N) ≥ D (33.11)
Applying the interactive response time law to the bounds specified by Equations (33.10) and (33.11), we get

and


FIGURE 33.3  Typical asymptotic bounds.

Combining these bounds with Equations (33.10) and (33.11), we get the asymptotic bounds as specified in Equations (33.8) and (33.9).

Figure 33.3 shows asymptotic bounds for a typical case. As shown in the figure, both throughput and response time bounds consist of two straight lines. The response time bounds consist of a horizontal straight line at R = D and a line passing through the point (-Z,O) at a slope of Dmax. The throughput bounds consist of a horizontal straight line at X = 1/Dmax. and a line passing through the origin at a slope of 1/(D + Z). The point of intersection of the two lines is called the knee. For both response time and throughput, the knee occurs at the same value of number of users. The number of jobs N* at the knee is given by

If the number of jobs is more than N*, then we can say with certainty that there is queueing somewhere in the system.


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