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# Limiting probabilities

Limiting probabilities. When do the limiting probabilities exist?. The limiting probabilities P j exist if (a) all states of the Markov chain communicate (i.e., starting in state i , there is a positive probability of ever being in state j , for all i , j and

## Limiting probabilities

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1. Limiting probabilities

2. When do the limiting probabilities exist? The limiting probabilities Pj exist if (a) all states of the Markov chain communicate (i.e., starting in state i, there is a positive probability of ever being in state j, for all i, j and (b) the Markov is positive recurrent (i.e, starting in any state, the mean time to return to that state is finite).

3. The M/M/1 queue l l l 0 1 2 3 m m m

4. The M/M/1 queue

5. The M/M/1 queue

6. The M/M/1 queue

7. The expected number in the system

8. The birth and death process l0 l1 l2 0 1 2 3 m1 m2 m3

9. The M/M/m queue

10. The M/M/m queue

11. The M/M/m queue

12. A machine repair model A system with M machines and one repairman. The time between machine is exponentially distributed with mean 1/l. Repair times are also exponentially distributed with mean 1/m. What is the average number of working machines? What is the fraction of time each machine is in use?

13. The machine repairman problem

14. The machine repairman problem

15. The machine repairman problem

16. The machine repairman problem

17. The machine repairman problem

18. The machine repairman problem

19. The machine repairman problem

20. The automated teller machine (ATM) problem Customers arrive to an ATM according to a Poisson process with rate l. If the customer finds more than N other customers at the machine, he/she does not wait and goes away. Machine transaction times are exponentially distributed with mean 1/m. What is the probability that a customer goes away? What is the average number of customers at the ATM? If the machine earns \$h per customer served, what is the average revenue the machine generates per unit time?

21. The M/M/1/N queue

22. The M/M/1/N queue

23. The M/M/1/N queue

24. The M/M/1/N queue

25. The production inventory problem Consider a production system that manufacturers a single product. Production times are exponentially distributed with mean 1/m. The production facility can produce ahead of demand by holding finished goods inventory. Orders from customers arrives according to a Poisson process with rate l. If there is inventory on-hand, the order is satisfied immediately. Otherwise, the order is backordered. The production system incurs a holding cost \$h per unit of held inventory per unit time and a backorder cost \$b per unit backordered per unit time. The production system manages its finished goods inventory using a base-stock policy with base-stock level s.

26. The production inventory problem • What is the expected inventory level? • What is expected backorder level? • What is the expected total cost? • What is the optimal base-stock level?

27. Three basic processes I: level of finished goods inventory B: number of backorders (backorder level) IO: inventory on order.

28. Three basic processes Under a base-stock policy, the arrival of each customer order triggers the placement of an order with the production system  s = I + IO – B  s = E[I] + E[IO] – E[B]

29. Three basic processes I and B cannot be positive at the same time  I = max(0, s - IO) = (s – IO)+ E[I] = E[(s – IO)+]  B = max(0, IO - s) = (IO - s)+  E[B] = E[(IO - s)+]

30. The production system behaves like an M/M/1 queue, with IO corresponding to the number of customers in the system.

31. Expected backorder level

32. Expected inventory level

33. Expected cost

34. Optimal base-stock level

35. Optimal base-stock level

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