Discrete-time stochastic processes

The service times in the two systems are independent

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Unformatted text preview: a multiprocessor computer facility, a job can be queued waiting for service at one processor, then go to wait for another processor, and so forth; frequently the same processor is visited several times before the job is completed. In a data network, packets traverse multiple intermediate nodes; at each node they enter a queue waiting for transmission to other nodes. Such systems are modeled by a network of queues, and Jackson networks are perhaps the simplest models of such networks. In such a model, we have a network of k interconnected queueing systems which we call nodes. Each of the k nodes receives customers (i.e., tasks or jobs) both from outside the network (exogenous inputs) and from other nodes within the network (endogenous inputs). It is assumed that the exogenous inputs to each node i form a Poisson process of rate ri and that these Poisson processes are independent of each other. For analytical convenience, we regard this as a single Poisson input process of rate ∏0 , with each input independently going to each node i with probability Q0i = ri /∏0i . ∏0 Q02 ❅ ❅✓✏ ❘ ❅ Q11 ✿ ✘1✐ ✒✑ °■ ° ✠ ° Q 10 Q13 Q12 Q21 Q31 Q32 ✓✏ ❘3✙ ✒✑ ✒ ° ° ❖❅ ❅ Q30 ❘ ❅ ° ∏0 Q03 °∏0 Q02 ° ✓✏ ✠ ° q 2 Q22 ② ✒✑ ✯❅ ❅ Q20 ❘ ❅ Q23 Q33 Figure 6.13: A Jackson network with 3 nodes. Given a departure from node i, the probability that departure goes to node j (or, for j = 0, departs the system) is Qij . Note that a departure from node i can re-enter node i with probability Qii . The overall exogenous arrival rate is ∏0 , and, conditional on an arrival, the probability the arrival enters node i is Q0i . Each node i contains a single server, and the successive service times at node i are IID random variables with an exponentially distributed service time of rate µi . The service times at each node are also independent of the service times at all other nodes and independent of the exogenous arrival times at all nodes. When a customer completes service at a given 260 CHAPTER 6. MARKOV PROCESSES WITH COUNTABLE STATE SPACES node i, that customer is routed to node j with probability Qij (see Figure 6.13). It is also possible for the customer to depart from the network entirely (called an exogenous P departure), and this occurs with probability Qi0 = 1 − j ≥1 Qij . For a customer departing from node i, the next node j is a random variable with PMF {Qij , 0 ≤ j ≤ k}. Successive choices of the next node for customers at node i are IID, independent of the customer routing at other nodes, independent of all service times, and independent of the exogenous inputs. Notationally, we are regarding the outside world as a fictitious node 0 from which customers appear and to which they disappear. When a customer is routed from node i to node j , it is assumed that the routing is instantaneous; thus at the epoch of a departure from node i, there is a simultaneous endogenous arrival at node j . Thus a node j receives Poisson exogenous arrivals from outside the system at rate ∏0 Q0j and receives endogenous arrivals from other nodes according to the probabilistic rules just described. We can visualize these combined exogenous and endogenous arrivals as being served in FCFS fashion, but it really makes no difference in which order they are served, since the customers are statistically identical and simply give rise to service at node j at rate µj whenever there are customers to be served. The Jackson queueing network, as just defined, is fully described by the exogenous input rate ∏0 , the service rates {µi }, and the routing probabilities {Qij ; 0 ≤ i, j ≤ k}. The network as a whole is a Markov process in which the state is a vector m = (m1 , m2 , . . . , mk ), where mi , 1 ≤ i ≤ k, is the number of customers at node i. State changes occur upon exogenous arrivals to the various nodes, exogenous departures from the various nodes, and departures from one node that enter another node. In a vanishingly small interval δ of time, given that the state at the beginning of that interval is m , an exogenous arrival at node j occurs in the interval with probability ∏0 Q0j δ and changes the state to m 0 = m + e j where e j is a unit vector with a one in position j . If mi > 0, an exogenous departure from node i occurs in the interval with probability µi Qi0 δ and changes the state to m 0 = m − e i . Finally, if mi > 0, a departure from node i entering node j occurs in the interval with probability µi Qij δ and changes the state to m 0 = m − e i + e j . Thus, the transition rates are given by qm ,m 0 = ∏0 Q0j = µi Qi0 = µi Qij =0 for m 0 = m + e j , 0 for m = m − m i , 1≤i≤k mi > 0, 0 (6.59) 1≤i≤k for m = m − e i + e j , mi > 0, 1 ≤ i, j ≤ k (6.60) (6.61) 0 for all other choices of m . Note that a departure from node i that re-enters node i causes a transition from state m back into state m ; we disallowed such transitions in sections 6.1 and 6.2, but s...
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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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