Discrete-time stochastic processes

94 695 696 697 n m xn d b using this hypothesis

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Unformatted text preview: onential servers. With this modification, the transition rates in (6.60) and (6.61) are modified by replacing µi with µi,mi . The hypothesized backward transition rates are modified in the same way, and the only effect of these changes is to replace ρi and ρj for each i and j in (6.72)-(6.74) with ρi,mi = ∏i /µi,mi and ρj,mj = ∏j /µj,mj . With this change, (6.75) becomes p(m ) = k Y pi (mi ) = i=1 pi (0) = 1 + k Y j =1 1 X m−1 Y m=1 j =0 pi (0) −1 ρi,j mi Y ρi,j (6.79) j =0 . (6.80) Thus, p(m ) is given by the product distribution of k individual birth death systems. 6.7.1 Closed Jackson networks The second generalization is to a network of queues with a fixed number M of customers in the system and with no exogenous inputs or outputs. Such networks are called closed Jackson networks, whereas the networks analyzed above are often called open Jackson networks. Suppose a k node closed network has routing probabilities Qij , 1 ≤ i, j ≤ k, P where j Qij = 1, and has exponential service times of rate µi (this can be generalized to µi,mi as above). We make the same assumptions as before about independence of service variables and routing variables, and assume that there is a path between each pair of nodes. Since {Qij ; 1 ≤ i, j ≤ k} forms an irreducible stochastic matrix, there is a one dimensional set of solutions to the steady state equations X ∏j = ∏i Qij ; 1 ≤ j ≤ k. (6.81) i We interpret ∏i as the time-average rate of transitions that go into node i. Since this set of equations can only be solved within an unknown multiplicative constant, and since this constant can only be determined at the end of the argument, we define {πi ; 1 ≤ i ≤ k} as the particular solution of (6.81) satisfying πj = X i πi Qij ; 1 ≤ j ≤ k; X πi = 1. (6.82) i Thus, for all i, ∏i = απi , where α is some unknown constant. The state of the Markov P process is again taken as m = (m1 , m2 , . . . , mk ) with the condition i mi = M . The transition rates of the Markov process are the same as for open networks, except that there are no exogenous arrivals or departures; thus (6.59)-(6.61) are replaced by qm ,m 0 = µi Qij for m 0 = m − e i + e j , mi > 0, 1 ≤ i, j ≤ k. (6.83) 6.7. JACKSON NETWORKS 265 We hypothesize that the backward time process is also a closed Jackson network, and as before, we conclude that if the hypothesis is true, the backward transition rates should be ∗ qm ,m 0 = µi Q∗ ij where ∏i Qij = ∏j Q∗i j for m 0 = m − e i + e j , mi > 0, 1 ≤ i, j ≤ k for 1 ≤ i, j ≤ k. (6.84) (6.85) In order to use Theorem 6.5 again, we must verify that a PMF p(m ) exists satisfying p(m )P ,m 0 = p(m 0 P∗m 0 ,m for all possible states and transitions, and we must also verify qm )q that m 0 qm ,m 0 = m 0 q ∗m ,m 0 for all possible m . This latter verification is virtually the same as before and is left as an exercise. The former verification, with the use of (72), (73), and (74), becomes p(m )(µi /∏i ) = p(m 0 )(µj /∏j ) for m 0 = m − e i + e j , mi > 0. (6.86) Using the open network solution to guide our intuition, we see that the following choice of P p(m ) satisfies (6.86) for all possible m (i.e., all m such that i mi = M ) p(m ) = A k Y (∏i /µi )mi ; for m such that i=1 X mi = M . (6.87) i The constant A is a normalizing constant, chosen to make p(m ) sum to unity. The problem with (6.87) is that we do not know ∏i (except within a multiplicative constant independent of i). Fortunately, however, if we substitute πi /α for ∏i , we see that α is raised to the power −M , independent of the state m . Thus, letting A0 = Aα − M , our solution becomes p(m ) = A0 K Y (πi /µi )mi ; for m such that i=1 1 A0 = m: X P k Y i=1 i mi =M √ πi µi !mi X mi = M . (6.88) i . (6.89) Note that the steady state distribution of the closed Jackson network has been found without solving for the time-average transition rates. Note also that the steady state distribution looks very similar to that for an open network; that is, it is a product distribution over the nodes with a geometric type distribution within each node. This is somewhat misleading, however, since the constant A0 can be quite difficult to calculate. It is surprising at first that the parameter of the geometric distribution can be changed by a constant multiplier in (6.88) and (6.89) (i.e., πi could be replaced with ∏i ) and the solution does not change; the important quantity is the relative values of πi /µi from one value of i to another rather than the absolute value. In order to find ∏i (and this is important, since it says how quickly the system is doing its work), note that ∏i = µi Pr {mi > 0}). Solving for Pr {mi > 0} requires finding the constant A0 in (6.83). In fact, the ma jor difference between open and closed networks is that the relevant constants for closed networks are tedious to calculate (even by computer) for large networks and large M . 2...
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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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