13.6 Open Models


Consider a distributed system environment, made up of a collection of clients and servers connected via a high speed local area network (LAN). Servers are placed in the system to provide service upon request. A file server is a key component of this kind of environment. Its purpose is to provide file services for the client processes on the network. Basically, a file server is composed of processors, memory, and a disk subsystem. The workload of a file server can be viewed as a series of file service requests such as read, write, create, and remove, which arrive from the workstations via the LAN. The operation of a file server can be viewed as follows. Requests arrive to the server. A typical request enters the server, possibly queues up while waiting for some resource (processor, memory, disks) to become available, obtains service from the resources, and exits the server. The number of requests being handled concurrently by the server varies over time depending on factors such as the load placed by the clients (i.e., the request arrival rate), the file system capacity, the available memory, and the processor speed.

A file server is a good example of a system suitable to be modeled as an open model. It exhibits the key characteristic of open systems: the variation of the number of customers over time. In practice, the number of customers (transactions, processes, requests) varies dynamically, due to such factors such as process termination, process initiation, and process spawning. An open class is able to represent this variation because it has a potentially unlimited number of customers. Chapter 10 introduces and analyzes the birth-death system, which gives the underlying theory for single-class open model. This analysis is extended here to multiclass models.

Figure 13.5. Approximate MVA algorithm for multiple classes.

graphics/13fig05.gif

The load intensity of a multiclass model with R open classes and K devices is represented by the vector graphics/362fig01.gif = (l1,l2, ···, lR), where lr indicates the arrival rate of class r customers. As illustrated in Figure 13.6, the goal of the analysis of multiclass open models is to determine, for each class, performance measures such as average response time, graphics/363fig01.gif, and queue lengths, graphics/363fig02.gif, as a function of the load intensity, graphics/362fig01.gif.

Figure 13.6. Multiple class open models.

graphics/13fig06.gif

13.6.1 Analysis of Multiclass Open Models

In steady state, the throughput of class r equals its arrival rate. Thus,

Equation 13.6.11

graphics/13equ611.gif


The application of Little's Law to each device gives

Equation 13.6.12

graphics/13equ612.gif


where Ri,r (graphics/362fig01.gif) is the average class r customer response time per visit to device i.

The average residence time for the entire execution is R'i,r = Vi,rRi,r. Using the Forced Flow Law and Eq. (13.6.11), the throughput of class r is

Equation 13.6.13

graphics/13equ613.gif


Using Eq. (13.6.13) in Eq. (13.6.12) the average queue length per device for each class becomes

Equation 13.6.14

graphics/13equ614.gif


Combining the Utilization Law and the Forced Flow Law, the utilization of device i by class r customers can be written as

Equation 13.6.15

graphics/13equ615.gif


Thus, to compute the average number of class r customers in service center i, graphics/364fig01.gif is needed as a function of the input parameters (i.e., graphics/362fig01.gif and the service demands Di,r's). The average time a class r customer spends at a device, from arrival until completion, has two components: the time for receiving service and the time spent in queue. The latter is equal to the time required to service customers that are currently in the device when the customer arrives. Thus,

Equation 13.6.16

graphics/13equ616.gif


where graphics/364fig02.gif is the average queue length at device i seen by an arriving class r customer when the load on the system is graphics/362fig01.gif For delay servers, graphics/364fig03.gif = Di,r.

The arrival theorem [17] states that in an open product-form queuing network, a class r arriving customer at service center i sees the steady-state distribution of the device state, which is given by the queue length. (Note: This is consistent with the arrival theorem result concerning closed systems. In closed systems, an arriving customer see the steady-state distribution with itself removed. In an open system, since there is an infinite customer population, removing oneself from the network has no effect. Thus, the steady-state distribution seen by an arriving customer is equal to the overall steady-state distribution.) Thus,

Equation 13.6.17

graphics/13equ617.gif


From Eqs. (13.6.16) and (13.6.17), we get

Equation 13.6.18

graphics/13equ618.gif


Substituting Eq. (13.6.18) into Eq. (13.6.14), yields

Equation 13.6.19

graphics/13equ619.gif


Notice that expression [1 + graphics/364fig04.gif in Eq. (13.6.19) does not depend on class r. As a consequence, for any two classes r and s, we have

Equation 13.6.20

graphics/13equ620.gif


Using Eq. (13.6.20) and considering the fact that graphics/365fig01.gif, Eq. (13.6.19) can be rewritten as

Equation 13.6.21

graphics/13equ621.gif


Applying Little's Law to Eq. (13.6.21), the average residence time for class r customers at device i is

Equation 13.6.22

graphics/13equ622.gif


The interaction among the open classes of a multiclass model is explicitly represented by the term graphics/365fig05.gif of Eq. (13.6.22), which corresponds to the total utilization of device i by all the classes in the model.

The analysis of a product-form model with multiple open classes begins with the constraint that graphics/365fig05.gif 1 for all devices of the network and proceeds with the formulas summarized in Figure 13.7. From Eq. (13.6.15), the stability condition for an open model is

Equation 13.6.23

graphics/13equ623.gif


or

Equation 13.6.24

graphics/13equ624.gif


13.6.2 Open Models: Case Study

Consider a distributed environment made up of a number of client diskless computers connected via a high-speed LAN to a file server, composed of a single processor and one large disk. The company is planning to double the number of client computers. Because the system performance is critically dependent on the file server, management wishes to assess the impact of the expansion before it is implemented. So, the first question to be answered is: What is the predicted performance of the file server if the number of diskless workstations doubles?

Following the modeling paradigm of Figure 10.1, the initial step is constructing the baseline model, which begins with the workload characterization. The workload to the file server consists of file service requests, which can be grouped into three classes: read, write, and all others. The latter comprises control requests of the network file system and other file service requests less used. During a period of one hour, the file server was monitored and the following measurement data were collected over one hour: 18,000 reads, 7,200 writes, 3,600 file service requests other than reads and writes, processor utilization at 32%, and disk utilization at 48%.

Figure 13.7. Formulas for models with multiple open classes.

graphics/13fig07.gif

The measurement data also provide resource utilization on a per-class basis, as shown in Table 13.8. Using the measurement data and the operational relationship (Di,r = Ui,r/X0,r), each request is characterized in terms of its service demands. Once Di,r has been calculated, it is possible to compute Vi,r using the disk service time provided by the manufacturer. For the processor, the parameter Vproc,r is calculated using the following expression that relates the visit ratio at the processor to the visit ratios at the I/O devices in a central server model

Table 13.8. File Server Workload Characteristics

Class

Arrival Rate

Processor

Disk

U (%)

V

S

D (sec)

U (%)

V

S

D (sec)

Read

5 req/sec

9

3

0.006

0.018

20

2

0.020

0.040

Write

2 req/sec

18

6

0.015

0.090

20

5

0.020

0.100

Others

1 req/sec

5

5

0.100

0.050

8

4

0.020

0.080

Equation 13.6.25

graphics/13equ625.gif


where devices 2 through K are the I/O devices. Table 13.8 summarizes the parameters that characterize the file server workload. Motivated by simplicity, the analyst in charge of the capacity planning project decided to construct a single-class model of the file server. The model constructed is an open model where only the file service components (processor and disk) are directly represented. It is assumed that the file server has enough memory so that no request queues for memory. The workstations are implicitly represented in the workload model by the file service requests generated by them. The larger the number of workstations, the larger the request arrival rate. The single-class model equivalent to the three-class model is obtained by calculating the aggregate demands.

graphics/367equ01.gif


By solving the model, the following residence times are obtained: graphics/367fig01.gif = 0.04/(1 0.32) = 0.059 and graphics/367fig02.gif = 0.06/(1 0.48) = 0.115. The sum gives an average request response time of 0.174 seconds.

To answer the "what if" question, it is necessary to change the baseline model to reflect the effects of doubling the number of workstations. Since this number is not directly specified in the input parameters, some assumptions are necessary. It is assumed that the new workstations will have the same usage as the installed ones. This means they will run the same group of applications and will generate file service requests that follow the current pattern of requests. Thus, by increasing the number of workstations by 100%, the request arrival rate, likewise, is assumed to increase by 100%. Letting lnew = 2 x 8 = 16, the model is re-solved to obtain the predicted measures.

graphics/368equ01.gif


Therefore, if the number of workstations were doubled, the file server disk would saturate and the average request response time would increase from 0.174 sec to 1.611 sec, an 825% increase! The model clearly indicates that the server would be bogged down and users would suffer with long response times at the workstations.

Now consider some possible alternatives to support system expansion without impairing service levels. The current system performance will be used as a basis for the comparison of the various alternatives under consideration. Each alternative is evaluated by considering the relative change in system performance from the baseline model.

  • Server caching scheme. A new version of the operating system that provides a cache for the server is available. According to the vendor, read and write cache hit ratios of 70% and 60%, respectively, can be achieved. To reduce the impact of unanticipated failures, every write operation to cache will also be applied to the disk. Thus, the server cache can then be modeled by reducing the visit ratio of read requests [i.e., graphics/368fig01.gif = (1 hit ratio)Vdisk,read]. To have a better understanding of the impact of the server cache on the performance of each class of the workload, a three-class model is solved using the predicted arrival rate and letting graphics/368fig02.gif = (1 0.7) x 0.04 = 0.012.

  • Client caching scheme. In this case, when a read is issued at the workstation, the data read are stored in a local buffer pool, called client cache [10]. Subsequent remote reads may find the requested data in the local cache. This reduces significantly the number of requests that go to the server. It is assumed that due to reliability reasons, a write operation always goes to the disk in the server after updating the client cache. The introduction of a client cache in the workstations of the environment can be modeled by reducing the arrival rate of read requests. Thus, graphics/369fig01.gif = (1 client hit ratio)lread. Assuming a client hit ratio of 70%, the new value for the parameter that represents the change is given by graphics/369fig01.gif = (1 0.7) x 10 = 3.

  • Upgrade in disk subsystem. The third alternative considered is to upgrade the storage subsystem and to install a second disk unit in the file server. This change is represented in the model by adding a third service center. The original disk service demand will be equally split between the two disks (i.e., graphics/369fig02.gif).

For each alternative, a three-class model is evaluated according to the formulas of Figure 13.7. The model inputs consist of parameters from Table 13.8 and those that were calculated to represent each specific change in the baseline model. Table 13.9 summarizes the performance measures obtained for the three alternatives. To obtain a rank of the alternatives in terms of performance, aggregate response time is first calculated for each alternative, which is the average of the individual classes weighted by the class throughput. Table 13.10 displays the aggregate response times as well as the relative change in the response time from the baseline model. The pure expansion refers to the alternative that just increases the number of workstations. All alternatives assume that the number of workstations will be doubled.

Table 13.9. Results of an Open Three-Class Model

Alternative

Processor

Disk

Extra Disk

U (%)

R' (sec)

U (%)

R' (sec)

U (%)

R' (sec)

• Server cache

      

Read

18

0.050

12

0.038

-

-

Write

36

0.250

40

0.313

-

-

Other

10

0.139

16

0.250

-

-

• Client cache

      

Read

5.4

0.037

12

0.125

-

-

Write

36

0.185

40

0.313

-

-

Other

10

0.103

16

0.250

-

-

• Disk upgrade

      

Read

18

0.050

20

0.038

20

0.038

Write

36

0.250

20

0.096

20

0.096

Other

10

0.139

8

0.077

8

0.077

From the file server perspective, the best alternative is the upgrade in the disk subsystem, because it considerably diminishes the demand on the original disk. The client cache, although seemingly worse than the server cache, may be the best alternative from the system viewpoint. The key issue in this alternative is the reduction in the number of times that a user process goes to the file server. This reduction implies less overhead of network access, which in many systems has a high cost.



Performance by Design. Computer Capacity Planning by Example
Performance by Design: Computer Capacity Planning By Example
ISBN: 0130906735
EAN: 2147483647
Year: 2003
Pages: 166

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