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M/M/1 Queues

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

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M/M/1 Queues

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  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 1 2 j-1 j j+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 r <1, • Therefore:

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

  5. M/M/1 Queues • Lq 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. • Wq is the expected waiting time in the queue. • W and Wq 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 1 2 s s-1 s+1 j j-1 j+1

  10. M/M/s Queues

  11. M/M/s Queues All servers are busy with probability This probability is used to find L,Lq, W, Wq The following table gives values of this probabilities for various values of r and s

  12. M/M/s Queues

  13. M/M/s queuing systemNeeded 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

  14. 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 .

  15. M/M/1/c Queues • The arrival rate is

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

  17. M/M/1/c Queues • We shall use Little’s formula to find W and Wq. 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.

  18. M/M/1/c Queues

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