## 11.4 The M/G/1 QueueA well-studied special case of the G/G/1 queue occurs when the interarrival times are exponentially distributed but the service time distribution is any arbitrary (i.e., general) distribution. This queue is called an M/G/1 queue. The basic result for an M/G/1 queue, known as the Pollaczek-Khintchine (P-K) formula [2] for the average waiting time W, is
where is the square of the coefficient of variation of the service time distribution. [Note: if C The other equations for M/G/1 follow directly from Eqs. (11.2.1)-(11.2.3):
## Example 11.3.Suppose that e-mail messages arrive at an e-mail server from a Poisson process at a rate of 1.2 messages per second. Also suppose that 30% of the messages are processed in 0.1 sec, 50% in 0.3 sec, and 20% in 2 sec. What is the average time E[S] it takes to process a message? What is the average time W a message waits in the queue to be processed? What is the average response time T of an e-mail message? What is the average number of messages N The average time to process a message is
The utilization of the e-mail server is r = l x E[S] = 1.2 messages/sec x 0.58 seconds/message = 0.696. The coefficient of variation of the processing time, C
Thus, the standard deviation s
The coefficient of variation C
## Example 11.4.What is the ratio between the average waiting time of an M/G/1 queue with exponentially distributed service times and the waiting time of an M/G/1 queue with constant service times? The coefficient of variation C
Thus, the time spent in the waiting line at an exponential server is on average twice the time spent in the waiting line of a constant speed server. Figure 11.2 shows various curves of the average response time versus utilization for an M/G/1 queue with E[S] = 1 and for four values of C ## Figure 11.2. Response time of an M/G/1 queue for various values of C |

Performance by Design: Computer Capacity Planning By Example

ISBN: 0130906735

EAN: 2147483647

EAN: 2147483647

Year: 2003

Pages: 166

Pages: 166

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