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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

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## Limiting probabilities

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**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).**The M/M/1 queue**l l l 0 1 2 3 m m m**The birth and death process**l0 l1 l2 0 1 2 3 m1 m2 m3**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?**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?**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.**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?**Three basic processes**I: level of finished goods inventory B: number of backorders (backorder level) IO: inventory on order.**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]**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)+]**The production system behaves like an M/M/1 queue, with IO**corresponding to the number of customers in the system.

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