Queuing Analysis Examples: Single-Server & Multi-Server Queues

1. Queuing Analysis

2. 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.

6. Single-Server Queue

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

8. Example Scenario

9. Multi-Server Queue

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

11. 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.

12. 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

13. 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

14. More FormulasList of Parameters

15. More Queuing Formulas