1. M/M/1 Queues Customers arrive according to a Poisson process with rate. There is only one server. Service time is exponential with rate . 0 12 j-1 jj+1...

2. M/M/1 Queues We let  = / , so From Balance equations: As the stationary probabilities must sum to 1, therefore:

3. M/M/1 Queues But for  <1, Therefore:

4. M/M/1 Queues L is the expected number of entities in the system.

5. M/M/1 Queues L q is the expected number of entities in the queue.

6. Little’s Formula W is the expected waiting time in the system this includes the time in the line and the time in service. W q is the expected waiting time in the queue. W and W q can be found using the Little’s formula. ( explain for the deterministic case )

7. M/M/1 queuing model Summary

8. M/M/s Queues There are s servers. Customers arrive according to a Poisson process with rate, Service time for each entity is exponential with rate . Let  = /s 

9. M/M/s Queues Thus 0 12 s-1 s s+1... j-1 j j+1...

10. M/M/s Queues

11. All servers are busy with probability This probability is used to find L,L q, W, W q The following table gives values of this probabilities for various values of  and s

13. M/M/s Queues

14. M/M/s queuing system Needed for steady state Steady state occurs only if the arrival rate is less than the maximum service rate of the system –Equivalent to traffic intensity  = /s  < 1 Maximum service rate of the system is number of servers times service rate per server

15. M/M/1/c Queues Customers arrive according to a Poisson process with rate. The system has a finite capacity of c customers including the one in service. There is only one server. Service times are exponential with rate .

16. M/M/1/c Queues The arrival rate is

17. M/M/1/c Queues L is the expected number of entities in the system.

18. M/M/1/c Queues We shall use Little’s formula to find W and W q. Note that: –Recall that was the arrival rate. –But if there are c entities in the system, any arrivals find the system full, cannot “arrive”. –So of the arrivals per time unit, some proportion are turned away. –  c is the probability of the system being full. –So (1-  c ) is the actual rate of arrivals.

19. M/M/1/c Queues