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 = X 8 = bits
average service time = Ts = / 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 / ( ) = 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 Formulas List of Parameters

15. More Queuing Formulas