Discrete-time stochastic processes

# For more general random walks with x 0 there is

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Unformatted text preview: Thus n + 1 starts service immediately and Wn+1 = 0. This is the case for customer 3 in the ﬁgure. These two cases can be combined in the single equation Wn+1 = max[Wn + Yn − Xn+1 , 0]; for n ≥ 0; W0 = 0. (7.4) Since Yn and Xn+1 are coupled together in this equation for each n, it is convenient to deﬁne Un+1 = Yn − Xn+1 . Note that {Ui ; i ≥ 1} is a sequence of IID random variables. From (7.4), Wn = max[Wn−1 + Un , 0], and iterating on this equation, Wn = max[max[Wn−2 +Un−1 , 0]+Un , 0] = max[(Wn−2 + Un−1 + Un ), Un , 0] = max[(Wn−3 +Un−2 +Un−1 +Un ), (Un−1 +Un ), Un , 0] = ··· ··· (7.5) 7.2. THE WAITING TIME IN A G/G/1 QUEUE: 283 s3 s2 ✛ Arrivals 0 ✛ ✛ x1 s1 ✛ x2 ✲✛ ✲✛ y0 w1 ✲ x3 w2 ✲✛ y1 ✲✛ y2 ✲ ✲ Departures ✲ ✛ x2 + w2 = w1 + y1✲ Figure 7.2: Sample path of arrivals and departures from a G/G/1 queue. Customer 0 arrives at time 0 and enters service immediately. Customer 1 arrives at time s1 = x1 . For the case shown above, customer 0 has not yet departed, i.e., x1 &lt; y0 , so customer 1’s time in queue is w1 = y0 − x1 . As illustrated, customer 1’s sytem time (queueing time plus service time) is w1 + y1 . Customer 2 arrives at s2 = x1 + x2 . For the case shown above, this is before customer 1 departs at y0 + y1 . Thus, customer 2’s wait in queue is w2 = y0 + y1 − x1 − x2 . As illustrated above, x2 + w2 is also equal to customer 1’s system time, so w2 = w1 + y1 − x2 . Customer 3 arrives when the system is empty, so it enters service immediately with no wait in queue, i.e., w3 = 0. = max[(U1 +U2 + . . . +Un ), (U2 +U3 + . . . +Un ), . . . , (Un−1 +Un ), Un , 0]. (7.6) It is not necessary for the theorem below, but we can understand this maximization better by realizing that if the maximization is achieved at Ui + Ui+1 + · · · + Un , then a busy period must start with the arrival of customer i − 1 and continue at least through the service of customer n. To see this intuitively, note that the analysis above starts with the arrival of customer 0 to an empty system at time 0, but the choice of 0 time and customer number 0 has nothing to do with the analysis, and thus the analysis is valid for any arrival to an empty system. Choosing the largest customer number before n that starts a busy period must then give the correct waiting time and thus maximize (7.5). Exercise 7.2 provides further insight into this maximization. n n Deﬁne Z1 = Un , deﬁne Z2 = Un + Un−1 , and in general, for i ≤ n, deﬁne Zin = Un + Un−1 + n · · · + Un−i+1 . Thus Zn = Un + · · · + U1 . With these deﬁnitions, (7.5) becomes n n n Wn = max[0, Z1 , Z2 , . . . , Zn ]. (7.7) Note that the terms in {Zin ; 1 ≤ i ≤ n} are the ﬁrst n terms of a random walk, but it is not the random walk based on U1 , U2 , . . . , but rather the random walk going backward, starting with Un . Note also that Wn+1 , for example, is the maximum of a diﬀerent set of variables, i.e., it is the walk going backward from Un+1 . Fortunately, this doesn’t matter for the analysis since the ordered variables (Un , Un−1 . . . , U1 ) are statistically identical to (U1 , . . . , Un ). The probability that the wait is greater than or equal to a given value α is n n n Pr {Wn ≥ α} = Pr {max[0, Z1 , Z2 , . . . , Zn ] ≥ α} . (7.8) This says that, for the nth customer, Pr {Wn ≥ α} is equal to the probability that the random walk {Zin ; 1 ≤ i ≤ n} crosses a threshold at α by the nth trial. Because of the 284 CHAPTER 7. RANDOM WALKS, LARGE DEVIATIONS, AND MARTINGALES initialization used in the analysis, we see that Wn is the waiting time in queue of the nth arrival after the beginning of a busy period (although this nth arrival might belong to a a later busy period than that initial busy period). As noted above, (Un , Un−1 , . . . , U1 ) is statistically identical to (U1 , . . . , Un ) and thus Pr {Wn ≥ α} is the same as the probability that the ﬁrst n terms of a random walk based on {Ui ; i ≥ 1} crosses a threshold at α. Since the ﬁrst n + 1 terms of this random walk provide one more opportunity to cross α than the ﬁrst n terms, we see that · · · ≤ Pr {Wn ≥ α} ≤ Pr {Wn+1 ≥ α} ≤ · · · ≤ 1. (7.9) Since this sequence of probabilities is non-decreasing, it must have a limit as n → 1, and this limit is denoted Pr {W ≥ α}. Mathematically,2 this limit is the probability that a random walk based on {Ui ; i ≥ 1} ever crosses a threshold at α. Physically, this limit is the probability that the waiting time in queue is at least α for any given very large-numbered customer (i.e., for customer n when the inﬂuence of a busy period starting n customers earlier has died out). These results are summarized in the following theorem. Theorem 7.1. Let {Xi ; i ≥ 1} be the interarrival intervals of a G/G/1 queue, let {Yi ; i ≥ 0} be the service times, and assume that the system is empty at time 0 when customer 0 arrives. Let Wn be the time that the nth cus...
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