Queuing Analysis 1 An example of a Queue

Queuing Analysis 1

An example of a Queue • Web server: handles requests in 1 msec • If requests arrive at a constant rate of 1000 req/sec or less, everything works fine. • In reality, arrival rate is not constant but varies. • Suppose arrival rate is irregular with an average of 500 req/sec. 2

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Single-Server Queue 6

Total time as a function of utilization • Theoretical maximum input rate that can be handled by the system is: • In practice: 70 -90%. 7

Example Scenario 8

Multi-Server Queue 9

A Multi-Server queue is more efficient than multiple Single-Server Queues, as shown below: 10

Example • Messages arrive at a switching center for a particular outgoing communication line in a Poisson manner with a mean arrival rate of 180 messages per hour. Message length is distributed exponentially with a mean length of 14, 400 characters. Line speed is 9600 bps. 11

Example (cont’d) • What is the mean waiting time in the switching center? mean message length = 14400 X 8 = 115200 bits average service time = Ts = 115200 / 9600 = 12 sec arrival rate = = 180 / 3600 = 0. 05 message/sec utilization = = 0. 05 x 12 = 0. 6 mean waiting time = T = 0. 6 x 12 / (1 - 0. 6) = 18 sec 12

Example (cont’d) How many messages will be waiting in the switching center for transmission on average? messages waiting = = 0. 6 x 0. 6/(1 - 0. 6) = 0. 9 messages 13

More Formulas List of Parameters 14

More Queuing Formulas 15