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3.4 Using QN Models

One of the most important aspects in using QN models to predict performance is to understand what models to use and how to obtain the data for the model. In Chapter 2, different types and uses of QN models (open, closed, single class, or multiclass) were discussed. Numerical examples that illustrate the process are provided here.

Example 3.12.

A Web server, composed of a single CPU and single disk, was monitored for one hour. The main workload of the server can be divided into HTML files and requests for image files. During the measurement interval 14,040 requests for HTML files and 1,034 requests for image files are processed. An analysis of the Web server log shows that HTML files are 3,000-bytes long and image files are 15,000-bytes long on average. The average disk service time is 12 msec for 1,000-byte blocks. The CPU demand, in seconds, per HTTP request, is given by the expression CPUDemand = 0.008 + 0.002 × RequestSize, where RequestSize is given in the number of 1000-byte blocks processed. This expression for the CPU demand indicates that there is a constant time associated to processing a request (i.e., 0.008 seconds) regardless of the size of the file being requested. This constant time involves opening a TCP connection, analyzing the HTTP request, and opening the requested file. The second component of the CPU demand is proportional to the file size since the CPU is involved in each I/O operation. What is the response time for HTML and image file requests for the current load and for a load five times larger?

Since the workload is characterized as being composed of two types of requests, a two-class queuing network model is required. Should an open or closed model be used? The answer depends on how the workload intensity is specified. In this example, the load is specified by the number of requests of each type processed during the measurement interval. In other words, the arrival rate for each type of requests is:

Equation 3.4.24


This workload intensity is constant and does not depend on a fixed number of customers. Therefore, an open QN model as described in Chapter 13 is chosen. The next step is to compute the service demands for the CPU and disk for HTML and image file requests. Using the expression for CPU time, the service demand for the CPU for HTML and image requests can be computed by using the corresponding file sizes in 1,000-byte blocks for each case as: DCPU,HTML = 0.008 + 0.002 × 3 = 0.014 sec and DCPU,image = 0.008 + 0.002 × 15 = 0.038 sec. The disk service demand is computed by multiplying the number of blocks read for each type of request by the service time per block. That is, Ddisk,HTML = 3×0.012 = 0.036 sec and Ddisk,image = 15 × 0.012 = 0.18 sec. By entering this data into the MS Excel OpenQN.XLS workbook that comes with this book and solving the model, the results in Table 3.3 are obtained.

Table 3.3. Service Demands, Arrival Rates, and Performance Metrics for Ex. 3.12




Arrival Rate (req/sec)



Service Demands (sec)







Utilizations (%)







Residence times (sec)







Response times (sec)



In the case of open models, the throughput is equal to the arrival rate. Consider what happens under a five-fold increase in the load. The arrival rates become λHTML = 5 × 3.9 = 19.5 requests/sec and λimage = 5 × 0.29 = 1.45 requests/sec. Solving the model with these values of the arrival rates, new response times of 0.93 sec for HTML and 4.61 sec for image requests are obtained. Thus, image file requests experience an increase in their response time by a factor of 17.5 and requests for HTML files experience a response time increased by a factor of 15.5. At the new load level, the disk utilization reaches 96% as indicated by the model, up from its previous 19.2% utilization (i.e., 14% + 5.2%). This indicates that the original system has excess capacity, but a five-fold load increase is nearing its maximum capacity. ▪

Example 3.13.

Reconsider the Web server of Example 3.12. What is the response time and throughput of HTML and image file requests when there is an average of 14 HTML requests and 6 image file requests being executed concurrently at all times?

In this case, the workload is specified by a number of concurrent requests in execution and not by an arrival rate. In this situation, a closed multiclass QN model (described in Chapter 13) is now appropriate. This model can be solved using the MS Excel workbook ClosedQN.XLS. The service demands are the same as in Example 3.12. Solving the model, RHTML = 0.72 sec, Rimage = 3.57 sec, XHTML = 19.3 requests/sec, and Ximage = 1.7 requests/sec. By comparing these results against these in Example 3.12, when the workload is increased five-fold, similar performance magnitudes are observed. ▪

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