Discrete-time stochastic processes

From 564 pj is also equal to limt1 pr x t j if the

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Unformatted text preview: ft-moving chain are independent and have probability ∏δ in each increment δ , we conclude that departures in the right-moving chain are similarly Bernoulli. r ✟✟ ✟✟ r r r r r ✟✟ ✟✟ r ✛δ ✲ r r r r ❍❍ ❍❍r r r r r r r r ✟r ✟ ✟ r✟ ✲ Arrivals r r ✟r ✟✟ ✲ ✟r ✟ Departures r ✟✟ r ✟✟❍❍❍ ✟ r✟ ❍State r ❍❍ ❍❍r r r ❍❍ ✛ ❍❍Arrivals ✛ r r ❍❍ ❍❍ Departures ❍❍r ❍❍r r Figure 5.6: Sample function of M/M/1 chain over a busy period and corresponding arrivals and departures for right and left-moving chains. Arrivals and departures are viewed as occurring between the sample times, and an arrival in the left-moving chain between time nδ and (n + 1)δ corresponds to a departure in the right-moving chain between (n + 1)δ and nδ . The fact that the departure process is Bernoulli with departure probability ∏δ in each increment is surprising. Note that the probability of a departure in the interval (nδ − δ, nδ ] is µδ conditional on Xn−1 ≥ 1 and is 0 conditional on Xn−1 = 0. Since Pr {Xn−1 ≥ 1} = 1 − Pr {Xn−1 = 0} = ρ, we see that the unconditional probability of a departure in the 5.6. ROUND-ROBIN AND PROCESSOR SHARING 217 interval (nδ − δ, nδ ] is ρµδ = ∏δ as asserted above. The fact that successive departures are independent is much harder to derive without using reversibility (see exercise 5.12). Property 2: In the original (right-moving) chain, arrivals in the time increments after nδ are independent of Xn . Thus, for the left-moving chain, arrivals in time increments to the left of nδ are independent of the state of the chain at nδ . From the correspondence between sample paths, however, a left chain arrival is a right chain departure, so that for the right-moving chain, departures in the time increments prior to nδ are independent of Xn , which is equivalent to saying that the state Xn is independent of the prior departures. This means that if one observes the departures prior to time nδ , one obtains no information about the state of the chain at nδ . This is again a surprising result. To make it seem more plausible, note that an unusually large number of departures in an interval from (n − m)δ to nδ indicates that a large number of customers were probably in the system at time (n − m)δ , but it doesn’t appear to say much (and in fact it says exactly nothing) about the number remaining at nδ . The following theorem summarizes these results. Theorem 5.8 (Burke’s theorem for sampled-time). Given an M/M/1 Markov chain in steady-state with ∏ < µ, a) the departure process is Bernoul li, b) the state Xn at any time nδ is independent of departures prior to nδ . The proof of Burke’s theorem above did not use the fact that the departure probability is the same for all states except state 0. Thus these results remain valid for any birth-death chain with Bernoulli arrivals that are independent of the current state (i.e., for which Pi,i+1 = ∏δ for all i ≥ 0). One important example of such a chain is the sampled time approximation to an M/M/m queue. Here there are m servers, and the probability of departure from state i in an increment δ is µiδ for i ≤ m and µmδ for i > m. For the states to be recurrent, and thus for a steady-state to exist, ∏ must be less than µm. Sub ject to this restriction, properties a) and b) above are valid for sampled-time M/M/m queues. 5.6 Round-robin and processor sharing Typical queueing systems have one or more servers who each serve customers in FCFS order, serving one customer completely while other customers wait. These typical systems have larger average delay than necessary. For example, if two customers with service requirements of 10 and 1 units respectively are waiting when a single server becomes empty, then serving the first before the second results in departures at times 10 and 11, for an average delay of 10.5. Serving the customers in the opposite order results in departures at times 1 and 11, for an average delay of 6. Supermarkets have recognized this for years and have special express checkout lines for customers with small service requirements. Giving priority to customers with small service requirements, however, has some disadvantages; first, customers with high service requirements can feel discriminated against, and 218 CHAPTER 5. COUNTABLE-STATE MARKOV CHAINS second, it is not always possible to determine the service requirements of customers before they are served. The following alternative to priorities is popular both in the computer and data network industries. When a processor in a computer system has many jobs to accomplish, it often serves these jobs on a time-shared basis, spending a small increment of time on one, then the next, and so forth. In data networks, particularly high-speed networks, messages are broken into small fixed-length packets, and then the packets from different messages can be transmitted...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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