Pakes, A.G. On the tails of waiting-time distributions. J. Appl. Probab. 12 (1975), 555--564.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....subexponential tail asymptotics. So far, this kind of problem has been investigated in various queueing and insurance risk settings. When the interarrival time sequence and the service time sequence are mutually independent and both consisting of i.i.d. random variables (i.e. GI GI input) Pakes [15] show the subexponential property for the stationary actual waiting time distribution. The corresponding risk model is studied in [10] where the subexponential asymptotics of the ruin probability is obtained. Due to the property on convolutions of subexponential distributions, it can be readily ....

....If F e is subexponential, then under Assumptions 1 and 2, F e (x) 4.3) By Lemma 2.1(ii) 4.3) says that, if F e is subexponential, then the stationary workload distribution is also subexponential under the stated assumptions. This gives the same result as in the literature (see, e.g. [2, 4, 10, 11, 15, 20]) where the single server queues and the corresponding risk models in various di#erent settings have been investigated. Theorem 1 is derived from the following lemma: Lemma 4.1 Suppose that F e is subexponential. Then, under Assumptions 1 and 2, we have the followings: i) For any k = 1, 2, ....

A.G. Pakes, "On the tails of waiting-time distributions," J. Appl. Probab., vol. 12, pp. 555--564, 1975.

.... X Y B] B i where K 2 is a positive constant, 1 if P[Y n = 0] P[Y n = 1] 1, and = 0, otherwise. Next, the preceding inequality and Lemma 2 from [17] imply P[B V B] 1f 6= 0g K 2 1 o (P[X B] Thus, which, by Pakes theorem [27] and X 2 IR results in This proves the result for lattice valued X and Y , with Y being bounded. Next, we use the technique from [17, pp.98 99] to extend the result to the general case of non lattice valued X, Y with Y unbounded. If Y is unbounded we can always choose a truncated ....

A.G. Pakes. On the tails of waiting-time distribution. J. Appl. Probab., 12:555-564, 1975.

....of source i are generally distributed with mean 1= i . The burst sizes of source i have distribution B i ( Delta) with mean fi i . Thus, the traffic intensity of source i is ae i = i fi i . Let B i be a stochastic variable with distribution B i ( Delta) The next result is immediate from [16]. i ( Delta) 2 S and ae i c, then i xg i xg as x 1: 2.2. On Off processes Here, sources generate traffic according to independent On Off processes, alternating between On and Off periods. The Off periods of source i are generally distributed with mean 1= i . The On periods of ....

Pakes, A.G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555-564.

....G(x) and G 1 (x) Then the following result on the waiting time distribution asymptotics of the GI=GI=1 queue holds (see Veraverbeke[36] Let K be the d.f. of A t . Theorem 3.5 (i) F 1 2S( K 1 2S and lim x 1 K(x) 1. ii) If K#K 1 2S,then (1. 11) This theorem was first proved in [32]# in [36] the same result was shown using a random walk technique. Some of the first applications of long tailed distributions in queueing theory were made by Cohen [13] and Borovkov[6] for functions of regular variations [26,5] Recent results on long tailed and subexponential asymptotics of a ....

A.G. Pakes. On the tails of waiting-time distribution. J. Appl. Probab., 12:555--564, 1975.

.... the modeling approach through self similar long range dependent processes, a more analytically tractable approach using fluid renewal type models in which renewal times are long tailed has been explored in [ANA95, HRS96] Queueing results in these two papers rely on the classical result by Pakes [PAK75] on the subexponential (long tailed) asymptotics of the waiting time distribution in a GI GI 1 queue or in earlier work of Cohen [COH73] which considered a regularly varying GI GI 1 queue. The result of Pakes has been generalized to a Markov modulated setting [AHK94, JLA95g] In [AHK94] the ....

A.G. Pakes, "On the tails of waiting-time distribution," J. Appl. Probab., vol. 12, pp. 555--564, 1975. References 36

....small enough, the rst term in the preceding sum, by Lemma 3.2 (ii) is upper bounded by C(P[B (1 2 )x] o(P[B x] where the last equality follows from Lemma 3.1 (i) The bound for the second term is as follows. By Pakes asymptotic result for the workload of a stable M G 1 queue [33] and assumption (3.4) dP[ B 0 6 u] o(P[B x] Next, by discretizing the last integral for some 0 one obtains Z ( j 1) x ( j )x B 0 6 u] o(P[B x] Q( 1 2 (j 1) x) P[ B 0 ( j )x] o(P[B x] Q( 1 2 (j 1) x) Q( j )x) o(P[B ....

A.G. Pakes. On the tails of waiting-time distribution. J. Appl. Probab., 12:555-564, 1975.

.... X, X i , i 1 is a sequence of i.i.d. r.v.s. The next sequence of lemmas will be used in Section 3 to estimate the deviations of process B. then for any # EX o(Q(x) n#1 # X n #n = 0. Proof: The lemma follows from stochastic dominance, Lemma 3 and Pakes theorem [15]. # xe Q(u) Proof: See the appendix. Lemma 6 Let X 0 a.s. and EX #. If N x = max n : i=1 X i x , then there exists # 0 such that for all x and 0 x EX u] Proof: See the appendix. 6 3 Main results In this section we present our ....

A.G. Pakes. On the tails of waiting-time distribution. J. Appl. Prob., 12:555--564, 1975.

....K be the stochastic variable representing the burst size. We assume that the burst size distribution K( Delta) is intermediately regularly varying with mean . The traffic intensity is ae = The following three results play a crucial role in the analysis in subsequent sections. Theorem II.1 (Pakes [10]) If K ( Delta) 2 S and ae c, then Theorem II.2 (Zwart [17] If K( Delta) 2 IR and ae c, then IP(P x) IP(K x(c Gamma ae) The above theorem immediately gives the tail distribution of the residual busy period. c, then x(c Gamma ae) B. On off processes ....

A.G. PAKES. On the tails of waiting-time distributions. Journal of Applied Probability 12, 555 -- 564, 1975.

....situation where the M G 1 process is neither asymptotically second order self similar nor longrange dependent. This shows that self similarity and or long range dependence are not necessary to produce heavy tailed queue lengths, a conclusion that can also be reached from an earlier result of Pakes [33] and Veravebeke [44] Since Proposition 4.1 holds for any distribution G it holds, in particular, for moderate tail distributions such as the Log normal and the Weibull distributions. Therefore, Proposition 4.1 tells us that the queue length will not be lighter than that of a moderate tail rv if ....

A. G. Pakes, \On the tails of waiting time distributions," J. of Appl. Prob., Vol. 12, pp. 555-564, 1975.

....have distribution function U 2 ( with mean 1 # 2 . The burst sizes B 2 have distribution function B 2 ( with mean # 2 #. Thus, the tra#c intensity is # 2 : # 2 # 2 . We assume that B 2 ( is regularly varying of index B 2 ( 2 for some # 2 1. The next result which is due to Pakes [17] then yields the tail behavior of the workload distribution of flow 2 in isolation. Theorem 2.1 If B and # 2 c, then 2 x . 5 Fluid input Here, flow 2 generates tra#c according to an On O# process, alternating between On and O# periods. The O# periods U 2 have distribution ....

Pakes, A.G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555--564.

....is chosen small enough, the first term in the preceding sum, by Lemma 1 (ii) is upper bounded by C(P[B ] the equality is implied by Lemma 9 (i) of the Appendix A. The bound for the second integral term is as follows. By Pakes asymptotic result for the workload of a stable M G 1 queue [39] and assumption (4) u] o(P[B x ] Next, by discretizing the last integral for some # 0 one obtains (## (j 1)#)x (## j#)x dP[ B 0 # u] e Q( 1 2## (j 1)#)x) P[ B 0 (# # j#)x e Q( 1 2## (j 1)#)x) Q( ## j#)x) where the last inequality is due ....

A.G. Pakes, "On the tails of waiting-time distribution," J. Appl. Probab., vol. 12, pp. 555--564, 1975.

....have distribution function U 2 ( with mean 1 # 2 . The burst sizes B 2 have distribution functionB 2 ( with mean # 2 #. Thus, the traffic intensity is # 2 : # 2 # 2 . We assume 2 ( is regularly varying of index 2 , i.e. B for some # 2 1. The next result which is due to Pakes [25] then yields the tail behavior of the workload distribution of flow 2 in isolation. Theorem II.1: and # 2 c, then 2 x . B.2 Fluid input Here, flow 2 generates traffic according to an On Off process, alternating between On and Off periods. The Off periods U 2 have distribution ....

Pakes, A.G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555--564.

....(2.15) Since 1 # f(s) can only have a zero for s 0, w 2 (s) in (2.14) is analytic at s = 0 too. 5 The Main Theorem We now establish asymptotics for the low priority waiting time ccdf W c 2 (t) Because of Theorem 2. 1, we can draw on previous theory in the time domain, just as Pakes [47] did to treat non exponential asymptotics for the GI G 1 FIFO model; see Athreya and Ney [20] Chover, Ney and Wainger [28] 29] and Abate et al. 11] 12] We also give alternative transform arguments, which are remarkably simple, but which either require extra conditions or which remain to be ....

....follows easily too. As in Conjectures 9.1 and 9.2, we conjecture that (12.17) holds for more general longtail distributions (when 0 is the rightmost singularity of g 1 (s) Remark 12.2. It is instructive to compare (12.17) with the corresponding result for the FIFO discipline, see [11] or Pakes [47], which is W c FIFO (t) # # 1 # G e (t) as t ## . 12.18) If G c (t) # t c , as t ##, then (12.17) is equivalent to W c 2 (t) # # (1 #) c 1 (1 # 1 ) c 1 t c 1 as t ## , 12.19) whereas (12.18) is equivalent to W c FIFO (t) # # (1 #) c 1 t c 1 as ....

A. G. Pakes, On the tails of waiting-time distributions, J. Appl. Prob. 12 (1975) 555--564.

....of these results to non renewal input processes is discussed. Asymptotics of queue size processes are also considered. 1 Introduction In the single station case, the asymptotical tail behaviour of stationary waiting and response time distributions is known; see, for example, 3] 11] 14] [19], 21] and [22] The multi station case has recently been studied in several papers. For instance, 7] considers the stationary waiting time and queue length distributions at each node in an acyclic queueing network, 1] deals with long range dependence in queueing networks with infinite server ....

....If F s 2 S and ae = IE=IE 1 , then IP i sup n1 n X k=1 ( k Gamma k ) x j ae 1 Gamma ae F s (x) Lemma 2. 15 is in essence due to Veraverbeke [23] Other proofs can also be found in [14] and [21] Earlier proofs based on more specific assumptions can also be found in [11] and [19]. A thorough treatment of subexponential distributions and their properties is given, for example, in [21] 3 Response Times in Tandem Networks We first show how the response times in tandem queues can be described by a set of recurrence equations involving only max and operations. This example ....

Pakes, A.G. (1975) On the tails of waiting-time distributions. J. Appl. Probab. 12, 555--564.

....:t 0 be canonical RBM (with drift 0 and variance 1) i.e. R(t) t B(t) 0st min s B(s) t 0 . 7) Let M n (k) 0 jk maxW n ( j ) k 0 , 8) and M(t) 0st maxR(s) t 0 . 9) Extreme value limits for M n (k) as k for any fixed n are given in Iglehart [12] and Pakes [16]. These limits require the extra condition E exp(eV k ) for some e 0 (10) and more, and involve relatively complicated normalization constants. However, it is natural to expect that the situation should simplify in heavy traffic. To start, we give the standard heavytraffic result in this ....

A. G. Pakes, "On the tails of waiting-time distributions," J. Appl. Prob., 12, 555-564 (1975).

....subexponential tail asymptotics. So far, this kind of problem has been investigated in various queueing and insurance risk settings. When the interarrival time sequence and the service time sequence are mutually independent and both consisting of i.i.d. random variables (i.e. GI GI input) Pakes [15] shows the subexponential property for the stationary actual waiting time distribution. The corresponding risk model is studied in [10] where the subexponential asymptotics of the ruin probability is obtained. Due to the property on convolutions of subexponential distributions, it can be readily ....

....Assumptions 1 and 2, lim x## P(V (0) x) F e (x) # 1 # . 4.3) By Lemma 2.1(ii) Theorem 1 says that, if F e is subexponential, then the stationary workload distribution is also subexponential under the imposed assumptions. 4. 3) gives the same result as in the literature (see, e.g. [2, 4, 10, 11, 15, 20]) where the single server queues and the corresponding risk models in various di#erent settings have been investigated. Theorem 1 is derived from the following lemma: Lemma 4.1 Suppose that F e is subexponential. Then, under Assumptions 1 and 2, we have the followings: i) For any k = 1, 2, ....

A.G. Pakes, "On the tails of waiting-time distributions," J. Appl. Probab., vol. 12, pp. 555--564, 1975.

....to derive a numerical solution for the waiting time distribution, even when the number of exponentials is very large. Moreover, we show that an asymptotic closed form expression for Pr(W t) prevails as N 1 and t 1. This expression is shown to coincide with a well known result of Pakes [18] (see also [7] as B 1. Using the notion of weak convergence [3] Feldmann and Whitt [9] proved that it is theoretically possible to approximate arbitrarily closely the waiting time distribution in a GI=G=1 queue with a Pareto service time distribution by the waiting distribution in a GI=G=1 ....

....that, as N 1, t 1 and B 1, B ( 1) 2 B ( 1) 2 ( N X n=0 B ( 1)n exp ( aB n t) at) 1 : 24) Combining Eq. 24) with Eq. 23) we obtain Pr(W t) a (1 ) 1) t 1 ; 25) as N 1, t 1 and B 1. This relation corresponds to the formula of Pakes [18] which states that the waiting time ccdf in a GI=G=1 queue satis es Eq. 25) when the service time has a power tail ccdf F (t) at) actually, Eq. 25) is only a special case of Pakes formula which applies also to more general subexponential distributions) 3.2. Numerical Results We ....

A. Pakes, On the Tails of Waiting-Time Distributions, J. Appl. Prob., Vol.12, pp.555-564, 1975.

No context found.

Pakes, A.G. On the tails of waiting-time distributions. J. Appl. Probab. 12 (1975), 555--564.

No context found.

A.G. Pakes, "On the tails of waiting-time distributions", Journal of Applied Probability, Vol.12:555-564, 1975.

No context found.

Pakes, A.G. (1975) On the tails of waiting time distributions. J. Appl. Probab. 12, 555--564.

No context found.

PAKES, A.G. (1975). On the tails of waiting-time distribution. J. Appl. Prob. 12, 555--564.

No context found.

A.G. Pakes. On the tails of waiting-time distribution. J. Appl. Probab., 12:555--564, 1975.

No context found.

PAKES, A. G. (1975). On the tails of waiting-time distribution. J. Appl. Prob. 12, 555--564.

No context found.

Pakes, A.G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555--564.

No context found.

Pakes, A.G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555--564.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC