HEAVY TRAFFIC LIMIT THEOREM FOR THE WORKLOAD PROCESS

present section we consider the workloadprocessfv_t;t0gof theGI=G=1 queue in heavy traf®c. In order to get a proper limiting process, we not only apply the same coef®cient of contraction D…r†, but we also scale time by a factor D₁…r†:ˆ …1 r†D…r†. It can be shown that

w…t;r†:ˆD…r†v_t=D1…r† …6:64†

converges in distribution forr"1, for everyt>0. It can further be shown that the thus scaled and contracted workload processfw…t;r†;t0g converges weakly to the workload process of a queueing model of which the input is described by a stable LeÂvy motion if 1<n<2 and by Brownian motion ifnˆ2 (withnthe index of the heaviest tail).

As a preparation for discussing these results in some detail, we consider thenoise traf®c ntˆkt rt and the workload backlog htˆkt t, with kt the amount of traf®c generated in‰0;t†. It is shown in Theorem 4.1 of Cohen [21] that nt andht, when scaled similarly asvt in Eq. (6.64), have limiting distributions forr!1. In that theorem, the same cases are considered as in Theorems 6.5.2±6.5.5Ðand for the coef®cient of contraction D…r† we take precisely the contraction factors of those theorems:

1. The tail ofB…t† is heavier than that ofA…t†:D…r† ˆD_B…r†. 2. The tail ofA…t† is heavier than that ofB…t†:D…r† ˆD_A…r†. 3. A…t†andB…t†have similar tails:D…r† ˆD_AB…r†.

De®ne

N…t;r†:ˆD…r†n_t=D1…r†; H…t;r†:ˆD…r†h_t=D1…r†: …6:65† Theorem 6.6.1

(i) The stochastic variables N…t;r† and H…t;r† converge in distribution for r"1.

(ii) Let N…t†and H…t†be stochastic variables with as distribution the limiting distribution of N…t;r†and H…t;r†,respectively;then fort0, Rerˆ0,

E‰e rN…t†=a_{Š ˆe}rn_t=a fornˆn_b<n_aorC_aˆ0; ˆern_t=a fornˆn_a<n_borC_bˆ0; ˆe‰dbrn_‡darn_Št=a fornˆnaˆnb; Ca>0; Cb>0; …6:66† E‰e rH…t†=a_{Š ˆe}tr=a_E‰e rN…t†=a_{Š: …6:67†}

Proof. See Cohen [21]. To give the reader insight into the approach, we sketch the proof for theM=G=1 case with heavy-tailed service time distribution as given in

Eq. (6.49) (see also Cohen [20]). This is a case withnˆn_b<n_a. In thisM=G=1 case, E‰e skt_{Š ˆ} P1 nˆ0e t=a…t=a†n n! bnfsg ˆe rst…1 bfsg†=bs; …6:68† so E‰e snt_{Š ˆe}rst‰1 …1 bfsg†=bsŠ_{: …6:69†} Hence, putting sˆrD…r†; tˆt=D₁…r†; …6:70†

it follows from Eqs. (6.65) and (6.69) that

E‰e rN…t;r†_{Š ˆexp} r 1 rrt 1 1 bfrD…r†g brD…r† : …6:71†

For the term in square brackets, we have the representation (6.49). It can be veri®ed thatg_b…arD…r††=…1 r† !0 forr"1. The fact thatD…r†is the unique zero of the contraction equation (6.55) with the property thatD…r† #0 forr"1 ®nally implies that

lim

r"1 E‰e

rN…t;r†=a_{Š ˆe}rn_t=a

: …6:72†

The convergence in distribution ofN…t;r†follows after application of the conver- gence theorem for Laplace±Stieltjes transforms [20]. The statements concerning

H…t;r†follow immediately from those forN…t;r†. j

REMARK 6.6.2. The distribution of the stochastic variable N…t†;t0, and

similarly of H…t†;t0, is a n-stable distribution (cf. Samorodnitsky and Taqqu [33, p. 5]). The process fN…t†;t0g is a process with stationary independent increments. As is evident from its LST representation, it is self-similar with index 1=n, that is,N…bt†andb1=n_{N…t†;b>0, have the same distribution for everyt>0.}

Note that fH…t†;t0g is not self-similar. In Samorodnitsky and Taqqu [33] this process fN…t†;t0g with 1<n<2 is called a n-stable LeÂvy motion with independent self-similar increments; fornˆ2, it is the Brownian motion.

In theM=G=1 case, it is veri®ed in Cohen [20] (using the fact that the process

fht;t0g has stationary independent increments) that the ®nite-dimensional distri- butions of the fN…t;r†;t0g process converge to those of the fN…t†;t0g

process.

The proof of Theorem 6.6.1 for theGI=G=1 case proceeds in principle along the same lines as sketched above for M=G=1. However, the proof of the weak convergence ofN…t;r†toN…t†is more complicated.

Let us now turn to the workload processes fv_t;t0g and the contracted and scaledfw…t;r†;t0g. We again discuss theM=G=1 case.

It follows from formula (4.99) of Cohen [15, p. 262] that, for Res0, Rey>0, …₁ tˆ0e yt_E‰e svtjv 0ˆ0Šdtˆ_{y s‰1 r}1_{1 bfsg†=bsŠ} 1 s …₁ tˆ0e yt_P‰v tˆ0jv0ˆ0Šdt …6:73†

with (cf. Formula (4.94) of Cohen [15, p. 262] and Eq. (6.29)) …₁ tˆ0e yt_P‰v tˆ0jv0ˆ0Šdtˆ1 _yr _{1 r‡}1_rE‰e yP_Šˆ 1 r y 1 E‰e yP…V†_Š: …6:74†

Having established this relation with P…V†, one can now use Theorem 6.3.7 that speci®es the heavy traf®c behavior of DP…r†P…V†; note that DP…r† ˆ

…1 r†D…r† ˆD1…r†. Again apply the transformation (6.70), with Rer0 and

0<1 r1, and also yˆoD1…r†; Reo0: Hence, …₁ tˆ0e yt_E‰e svtjv 0ˆ0Šdtˆ_D1 1…r† …₁ tˆ0e ot_E‰e rw…t;r†_{jw…0;r† ˆ0Šdt: …6:75†}

It follows (cf. Cohen [20]) that w…t;r† converges in distribution to a stochastic variable w_t for r"1 for every t0, and that the distribution of w_t, t0 is speci®ed as follows. For Reo>0, Rer0,

…₁ tˆ0e ot_E‰e rwt=a_jw 0ˆ0Šdtˆ_{o r…1}1_‡rn 1_† 1 r o 1 p…o† : …6:76† As a by-product we ®nd that lim t!1 E‰e rwt=a_{Š ˆ} 1 1‡rn 1; …6:77†

For theGI=G=1 queue, the workload processfv_t;t0g is described by Reich's formula ([15, p. 170]; in the sequel we assume that v₀ˆ0, in order to make the following derivations somewhat simpler):

v_tˆmax h_t; sup

0<u<t…ht hu†

; t0: …6:78†

It easily follows from Eqs. (6.64) and (6.78), withw…0;r† ˆ0, that

w…t;r† ˆmax H…t;r†; sup

0<u<t…H…t;r† H…u;r††

; t0: …6:79†

Let us also consider the M~=G~=1 queue with input process fN…t†;t0g and workloadw~t at timetde®ned by (cf. Reich's formula)

~

wtˆmax H…t†; sup

0<u<t…H…t† H…u††

; t0: …6:80†

It is shown in Cohen [20] for 1<n_b2 that the processfw…t;r†;t0g indeed converges weakly (in the Skorokhod topology) forr!1 to the processfw~_t;t0g. The proof is based on a theorem of Skorokhod for processes with independent increments [35]. Actually, the processfH…t;r†;t0gis a process with independent increments, of which for r"1 all the ®nite-dimensional distributions of its increments converge weakly to those of thefH…t†;t0gprocess. The latter process is itself a process with independent increments. Because sup_0<u<t…H…t;r† H…u;r†† is a continuous functional of the H…t;r† process, Skorokhod's theorem implies that the process fw…t;r†;t>0g converges weakly to the process

fw~t;t0g. In view of the conclusion below Eq. (6.75),wt'w~tfor 1<nb <2. So for theM=G=1 queue with service time distribution speci®ed by Eq. (6.49), we conclude the following. The workloadw…t;r†, contracted and scaled as in Eq. (6.64), converges in distribution forr"1 and everyt>0. The limiting distribution is speci®ed by Eq. (6.76). The limiting process is also speci®ed as the process that satis®es Reich's formula (6.80) for anM~=G~=1 queue with as input then-stable LeÂvy motionfN…t†;t0g. These results are generalizations of the diffusion approxima- tion of the M=G=1 queue with a ®nite service time variance. Similar results are obtained for theGI=G=1 queue, in particular, concerning the convergence of the input process fN…t;r†;t0g to n-stable LeÂvy motion fN…t†;t0g and of the

fw…t;r†;t0g process to the fw~_t;t0g process for r"1 (see Cohen [21]); however, the proofs become more complicated.

REMARK6.6.3. Although in this chapter only the GI=G=1 queue with instanta-

neous arrivals has been considered, many of the heavy traf®c results should have a counterpart in ¯uid queues with regularly varying active and=or silent period distributions. Indeed, several authors [7, 18, 24] have established explicit distribu- tional relations between buffer contents in ¯uid queues and (actual or virtual)

waiting times in theGI=G=1 queue (cf. also Remark 6.2.6). And even if such results are not at hand, it is intuitively clear that in heavy traf®c it does not play a crucial role whether work increases instantaneously or linearly.

6.7 CONCLUSION

This chapter has been devoted to the GI=G=1 queue with regularly varying interarrival and=or service time distribution.

First, we studied the tail behavior of the waiting time distribution in theM=G=1 case, for three different service disciplines: FCFS, LCFS preemptive resume, and processor sharing. The effect of the service discipline on the tail behavior was shown to be very pronounced, FCFS leading to heavier waiting time tails than the other two disciplines.

Second, we studied the heavy traf®c behavior of the waiting time, for the cases in which the variance of the interarrival and=or service time is in®nite. A coef®cient of contraction D…r† was identi®ed, such that the contracted waiting time D…r†W

converges in distribution when the traf®c loadr"1.

Third, the heavy traf®c behavior of the workload process fv_t;t0g was investigated. Both a coef®cient of contraction D…r† and a time-scaling factor D₁…r† were identi®ed, such that D…r†v_t=D1…r† converges in distribution for r"1, for allt0. The thus scaled and contracted workload process converges weakly to the workload process of a queue of which the input process is described by an-stable LeÂvy motion.

Some topics of our recent and present research on these three subjects are:

Investigation of the in¯uence of long-tailed traf®c characteristics of one type of customer on performance measures of other types of customer. Preliminary results are obtained in Boxma et al. [11] for the M=G=1 queue with (non)preemptive priority, in Zwart [41] for theM=G=1 queue with processor sharing and several customer classes, and in Borst et al. [6] for the generalized processor sharing discipline. The sometimes dramatic impact of queue scheduling disciplines on performance may have important implications for the choice of scheduling disciplines in designing switches in communication networks.

We have extended the heavy traf®c limit theorem 6.3.1 to the case of an

M=G=1 queue with priority classes, and we have exploited the resulting limit theorem to derive a heavy traf®c approximation for the waiting time distribu- tion of low-priority customers [11]. In Boxma and Cohen [9] we had already obtained an approximation for the waiting time distribution for theM=G=1 queue with heavy-tailed service time distribution by using the heavy traf®c limit theorem. In both studies, the resulting approximation is remarkably sharp, even when traf®c is not heavy at all. It is conjectured that similar approxima- tions can be developed for the GI=G=1 queue with heavy-tailed interarrival

and=or service time distribution. This is a point for further study. It is important to perform many more numerical experiments to get more insight into the effect of long-tailed traf®c characteristics on performance measures, and to be able to develop useful approximations.

The weak convergence results of Section 6.6 for the GI=G=1 workload in heavy traf®c can probably be extended in several directions. For example, it is of interest to studynetworksof queues.

ACKNOWLEDGMENT

The authors are indebted to Dr. V. Dumas and Professor A. J. Stam for interesting suggestions and discussions.

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