Modified two-moment approach forconstant-service tandem queues

A. Abo Taleb. M.Sc. and H.T. Mouftah, Ph.D., Sen.Mem.I.E.E.E.

Indexing terms: Networks, Mathematical techniques

Abstract: A modified two-moment (MTM) method is proposed for waiting-time analysis of constant-servicetandem queueing systems. Input to an intermediate queueing station consists of internal stream(s) provided bythe output from previous queue(s) in addition to external Poisson traffic. Internal traffic at a queue is assumedto have initially originated from external Poisson sources and is permitted to exit from the tandem path withfixed independent probability. Analysis of such complex configuration is not tractable and known exact resultsare restricted to very special cases. It is shown that the MTM method provides a systematic and more accuratetool for waiting-time analysis of such queueing structures as compared to other known approximations.

1 Introduction

Constant service queues are the building blocks used tomodel packet-switching communication networks in whichfixed-length packets flow. In this paper we consider atandem connection of N such queues, each of which maybe represented as shown in Fig. 1, where Xu is the average

I,

D-AP,

Fig. 1 Basic queueing model

arrival rate in packets/s on the jth input stream at the ithqueue, /, is the transmission time in seconds for packets onthe output channel of the ith queue and p{ is a fixed inde-pendent probability that the output of the ith queue pro-ceeds to the next queue. It is assumed that Xtl = Xl2 = 0,Xn is provided by the output of the preceding queue, Xi2 isderived from a separate M/D/l queue, or more generallyfrom a tandem path other than that under investigationand Xi3 is a Poisson external input. The case of more thantwo internal streams is a direct extension, and hence twointernal streams will be used to illustrate the proposedtechnique.

An exact solution is known for the case where no trafficis permitted to interfere with, or leave, the tandem path(i.e. Xjj = 0, i > 1,7 > 1, P, = 1) between source and desti-nation nodes [1, 2]. An exact solution has been alsoobtained for the case where N = 2, X22 = 0 and P, = 1provided that /x < /2 [3]. Apart from these special cases asolution for the general model proposed is not mathemati-cally tractable because of its non-Markovian nature.However, good approximations may be obtained underreasonable simplifying assumptions. The independenceassumption [4] is not a good candidate to invoke here asit is known to lead to a loose upper bound on the averagewaiting time [5]. In the absence of this assumption, Rubin[5] derived approximate formulas for N = 2, with internaltraffic, derived from an M/D/l queue. The approximationis simple, however, it may tend to overestimate the averagewaiting time at high loads. The impact of independencehas been reduced considerably in Reference 6 through theuse of a two-moment (TM) approach, in which departure

Paper 3818E (Cl, C2), first received 21st June 1984 and in revised form 29thJanuary 1985

The authors are with the Department of Electrical Engineering, Queens University,Kingston, Ontario K7L 3N6, Canada

and arrival processes are assumed renewal and consideredwith respect to their first two moments only. However, theTM technique does not perform adequately when mes-sages are of fixed length as it completely disregards thedeterministic component inherent in traffic.

In this paper a modified two-moment (MTM) method isproposed and shown to perform better than both Rubin'sand TM methods as it provides tighter bound on simula-tion results. In Section 2 a model for the probabilitydensity function (PDF) of interdepartures from a GI/D/lqueue is developed and used in Section 3 to describe thecomposite input process to a queue. Exact waiting-timeanalysis with respect to the assumptions made is presentedin Section 4. In Section 5 a method is introduced for com-puting the average waiting time for an arbitrary packet inboth the composite internal stream and the externalPoisson stream.

2 Output process

If the output channel of a GI/G/l queue has a utilisationp < 1, the first moment of the PDF of interdeparture timeswill be equal to the first moment of interarrival times dueto the conservation of flow. The second moment of interd-eparture times can be computed exactly if the exact valuesof the average waiting time and the coefficients of variationof both the arrival and service-time processes can be com-puted as indicated in Appendix 9. To use this information,the assumption of renewal arrivals and departures isretained after modifying the input process in the way dis-cussed in the next Section. The first two moments are thenfitted to the PDF a(t) given by

a(t) = gS(t-l) U)where / is the service time of the queue output channel,<5( •) and U( •) are the Dirac delta and unit step functions,respectively, and g, gt and g2 are the fitting constants.Conditions under which the fitting constants are non-negative are discussed in Appendix 9.

The PDF a(t) in eqn. 1 has the property that if the inputprocess is Poisson, it can readily be shown that a{t)becomes exactly the marginal PDF of interdepartures fromthe M/D/l queue [8]. In the MTM method, a{t) will beused to represent interdeparture times of the GI/D/lqueues in tandem when all external arrivals are assumed tobe Poisson. It is worth noting that the TM method [6],which will be compared in Section 6 with the MTMmethod, uses a hypo-hyperexponential distribution tocarry out the fitting. In the case where a portion of thetraffic is permitted to exit from the tandem path with fixed

170 IEE PROCEEDINGS, Vol. 132, Pt. E, No. 3, MAY 1985

independent probability, the first two moments should bemodified by using the thinning concept of renewal pro-cesses [9].

3 Composite input process

Input streams to a queue are assumed to be mutually inde-pendent. The composite input stream is obtained by usingthe law of superposition of renewal processes [10]. Priorto superposition, the PDF of interarrivals for the individ-ual internal streams is modified by appropriately shiftingits impulse and keeping its first two moments unaltered.Let / and I be the impulse positions before and after theshift, respectively, and /,• be the service time at the queueoutput channel. We then set

(2)

where A is the sum of all traffic rates in the compositeinput excluding that of the internal stream being shifted.The exponential term in eqn. 2 may be interpreted as thePoisson probability of a stream observing no arrivals fromthe other input streams during a service time period. It isseen from eqn. 2 that we always have /</,•; this is toensure that the composite PDF reflects the fact that aslong as there is a deterministic traffic component, interarri-val times, smaller than the deterministic ones due to inter-ference, will frequently be obtained. Emphasising thedeterministic interarrivals before superposition in this waycompensates for the assumption that the compositeprocess is renewable.

4 Waiting time analysis

Exact waiting-time analysis, with respect to the assumedrenewal arrival pattern of the previous Section, amounts tosolving the relevant Lindley's integral equation [11]. A sol-ution is obtained via the Wiener-Hopf technique [12],which yields the Laplace-Stieltjes transform (LST) W(s) ofthe waiting-time distribution. The mechanism of themethod is as follows: first form

c(s) = /4,( -s)e~sli (3)

where At(s) is the LST of the composite input PDF ofinterarrivals at the ith queue; next construct

1c(s) -

N(s)

D{s)(4)

where N(s) is a polynomial in s and D(s) is nonlinear in sdue to the exponential term appearing in eqn. 3 and theimpulses present in the PDFs of interarrivals. W(s) can beextracted from eqn. 4 to within a multiplication constantby removing the poles and zeros with Re {s} > 0. It is gen-erally difficult to factorise Laplace transforms which arenot rational. However, the careful choice of the PDF ineqn. 1 has been found to lead to D(s) being free from rootsin the RHP, and thus it need not be factorised. Althoughthe form of D(s) depends on the configuration under inves-tigation it normally contains terms such as

e(l~'i)s (5)

This term tends to infinity as s—• oo even after successivedifferentiation, unless I < /,• which is always guaranteed byrelation 2. The extracted PDF was further checked indi-rectly by several numerical inversions which alwaysresulted in a monotonic nondecreasing function of time.

Moreover, all the information drawn from W(s) thusobtained (e.g. probability of zero waiting time, averagewaiting time etc.) was consistent. The multiplication con-stant in W(s) is obtained from the final value theorem

Lim sW(s)= 1 (6)s - 0

The probability of zero waiting time, Po, is obtained fromthe initial value theorem

Po = Lim sW(s)s-*oo

The average waiting time W is obtained from

d sW(s)W = - Lim

s-0 ds

(7)

(8)

In evaluating the limits in eqns. 6-8 use is made ofL'Hospital's rule more than once, depending on thenumber of input streams involved.

5 Average waiting times for individual classes ofarrivals

Approximation methods [5, 6] solve for the averagewaiting time of an arbitrary packet in the composite inputstream. Exact analysis of the special case in Reference 3,referred to earlier, yielded the average waiting time ofpackets in both the internal and the composite inputstreams. It is also suggested that the waiting time ofpackets in the Poisson stream may be equal to the virtualwaiting time in the queue. This can be shown to be true forthe average of these quantities through the use of thegeneral conservation law developed in Reference 13. Forthe G/D/l queue with two classes of arrivals, the conserva-tion law can be stated as

V = p{w + 1/2)

and

= PlWl+p2W2

(9)

(10)

where V is the average virtual waiting time, p is the serverutilisation, p, is the server utilisation for the ith class andWt is the average waiting time of the /th class. Based on theassumption that Poisson arrivals normally see time aver-ages, one may use eqns. 8-10 to compute the averagewaiting times for two classes of arrival patterns when oneclass is Poisson, by taking V in eqn. 9 equal to the averagewaiting time of the Poisson class. The other class may be asingle internal stream or the superposition of more thanone internal stream. The validity of this approach is illus-trated in the examples of the next Section.

6 Results

In this Section the accuracy of the MTM approximationpresented above is evaluated. The evaluation is based onthe two tandem queues examples of Figs. 2 and 3. Theparameters of interest are the average waiting times in thesecond and third queues of Fig. 2 and the average waitingtimes in the individual input streams at the second queueof Fig. 3. The average waiting time in a queue is plotted

Fig. 2 Example 1 queueing model

IEE PROCEEDINGS, Vol. 132, Pt. E, No. 3, MAY 1985 171

against the utilisation of its outgoing channel, with the uti-lisation being varied by varying the channel service time.

12

o 0-

^23

Fig. 3 Example 2 queueing model

MTM

RBN

0 0.12 0.24 0.36 0.48 0.60 0.72 0.84utilisation

Fig. 4A Average waiting time in the second queue of Fig. 2: Compari-son with the M/D/l and Rabin's techniques

A, 3 = 0.1, /12 3 = 0.02 and/ , = 5

9

8

«. 7

.§ 6

| 5

5 4&

MTM

0 0.12 0.24 0.36 0.48 0.60 0.72 0.84utilisation

Fig. 4B Average waiting time in the second queue of Fig. 2: Compari-son with the two moment techniqueA13 = 0.1, / „ = 0.02 and /, = 5

.£ 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig. 4C Average waiting time in the third queue of Fig. 2

Xt3 = 0.1, X23 = 0.02, Xi3 = 0.02, /, = 5, and l2 = 1

172

0.8

Simulation curves (SIM) in Figs. 4A and B are due toRubin [5]. Other simulations are shown with their 95%confidence levels. It is clear from all the Figures that thePoisson assumption (MDl curve) for interarrivals at therespective queues is quite inadequate, as it loosely overesti-mates the average waiting time. Rubin's approximation isshown in Fig. 4A (RBN curve) to approach the MDl curveat higher loads much faster than the MTM curve. Com-parison with the ordinary TM method appears in Figs. 4B,C, 5A, B and C. From the latter Figures it is seen that theTM approach performs much better than the MDlassumption; however, it generally tends to overestimatethe average waiting time more than the MTM curve does.Fig. 5D is meant to confirm the fact that the traffic with adeterministic component always experiences less waitingtimes than the random traffic (see Reference 14 and refer-ences therein).

A more complex queueing model appears in Fig. 6. Theaverage waiting time at the output queue of this model is

10

Eg! 3a

MTM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9utilisation

Fig. 5A Average waiting time of internal traffic at the second queue ofFig. 3A13 = 0.15, A21 = 0.1, A23 = 0.05 and /, = 5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8utilisation

Fig. 5B Average waiting time for Poisson traffic at the second queue ofFig. 3Xl3 = 0.15, X2l = 0.1, / 2 3 = 0.05 and /, = 5

1EE PROCEEDINGS, Vol. 132, Pt. E, No. 3, MA Y 1985

shown in Fig. 7, where we see that the TM and MTMmethods exhibit the same behaviour noted in the previousexamples. Note that, in all examples, the ratio of internaltraffic to external traffic was chosen such that the internalcomponent is the higher in weight. It is expected, of course,

that, as the ratio of the Poisson component at a nodebecomes more dominant, the results get closer to those of

0.1 0.2 0.3 0A 0.5 0.6 0.7 0.8utilisation

Fig. 5C Average waiting time for an arbitrary packet at the secondqueue of Fig. 3

/ 1 3 = 0.15, A21 = 0.1, A23 = 0.05 and /, = 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9utilisation

Fig. 5D Average waiting times at the second queue of Fig. 3 usingMTM technique

A,j = 0.15, A21 = 0.1, x23 = 0.05 and /, = 5

a Poisson streamb Arbitrary packetc Internal stream

^33

o-\22

D-

Fig. 6 Example 3 queueing model

IEE PROCEEDINGS, Vol. 132, Pt. E, No. 3, MAY 1985

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig. 7 Average waiting time in the output queue of Fig. 6I, = U = /s = 5, l2 = 1, / l 3 = A22 = A32 = 0.1, A3 3 = 0.02

the M/D/l queue, which is an exact limiting case for all themethods compared.

7 Conclusion

An efficient method for waiting-time analysis of constant-service tandem queues in the absence of the independenceassumption has been developed. The method is a modifiedversion of the ordinary two-moment approach. A specificPDF for interdeparture times is adopted and adjusted toaccount for traffic interference. Exact waiting-time analysisis carried out to minimise the approximation phase of thetechnique. Simulation results confirm the validity of themethod and show that it compares favourably with theother known methods.

8 References

1 FRIEDMAN, H.D.: 'Reduction methods for tandem queueingsystems', Oper. Res., 1965, 13, pp. 121-131

2 RUBIN, I.: 'Path delays in communication networks', Appl. Math. &Optimiz., 1975, 1, (3), pp. 193-221

3 KAPLAN, M.: 'A two-fold tandem net with deterministic links andsource interference', Oper. Res., 1980, 28, (3), pp. 512-526

4 KLEINROCK, L.: 'Communication nets: Stochastic message flowand delay' (New York, McGraw-Hill, 1964)

5 RUBIN, I.: 'An approximate time-delay analysis for packet switchingcommunication networks', IEEE Trans., 1976, COM-24, pp. 210-222

6 KUEHN, P.J.: 'Approximate analysis of general queueing networksby decomposition', ibid., 1979, COM-27, pp. 113-126

7 MARSHAL, K.T.: 'Some relations between the distributions ofwaiting time, idle time and interoutput time in the G//G/1 queue',SI AM J., Appl. Math., 1968,16, (2), pp. 324-327

8 PACK, CD. : 'The output of an M/D/l queue', Oper. Res., 1975, 23,(4), pp. 750-760

9 GNEDENKO, B.V, and KOVALENKO, I.N.: 'Introduction toqueueing theory' (Wiener Bindery Ltd., Jerusalem, 1968)

10 COX, D.R.: 'Renewal theory' (Methuen, 1962)11 KLEINROCK, L.: 'Queueing systems, Vol. 1' (Wiley, New York,

1975)12 TITCHMARSH, E.C.: 'Introduction to the theory of Fourier integ-

rals' (O.U.P., 1948)13 SCHRAGE, L.: 'An alternative proof of a conservation law for the

queue G/G/V, Oper. Res., 1970,18, pp. 185-18714 WHITT, W.: 'Minimizing delays in the G//G/1 queue', ibid., 1984, 32,

(1), pp. 41-5115 ABO-TALEB, A, and MOUFTAH, H.T.: 'Performance of constant

service queueing systems'. Proceedings of IEEE Global Telecommuni-cation Conference, Atlanta, Nov. 1984, pp. 826-830

9 Appendix

For p < 1 the first moment mg of eqn. 1 should be equal tothe reciprocal of X, the average arrival rate. Let Cg be thecoefficient of variation of eqn. 1, then it is easy to see that

173

g, #! and g2 should satisfy

2(1 -P)2

0 = 1 - Cl + (1 - P):

2/1(1-p)

(11)

(12)

(13)

Relation 12 indicates that g^ is always positive, whileCg > 0 is sufficient to keep #2 positive. Relation 11 showsthat nonnegative g requires

Cl > (1 - p)2 (14)

Thus eqn. 14 is sufficient to keep both g and g2 non-negative. Let CD be the coefficient of variation of interde-partures. It is necessary to see when CD satisfies eqn. 14.Intuitively, CD assumes its smallest value in the absence ofinterference. Hence, we consider the simple case of the twoqueues Qo and Q shown in Fig. 8, where /0 and / are the

D-

Fig. 8 Noninterfered tandem path

service times at Qo and Q, respectively. Since Q is assumedGI/D/l we can write [7]

where CA, CD and Cs are the coefficients of variation ofinterarrival, interdeparture and service times, respectively,associated with Q and p = XL With deterministic servicetime Cs = 0, and therefore

(16)

(17)

-p)W

Using the waiting-time analysis of Section 4 we find

0 / „ > /

Comparing eqn. 20 with eqn. 14 we see that

C2D > (1 - p)2 / „< / (22)

so that the fit is always possible in this case. For I < l0,condition 14 is satisfied if

l - p 2 > ( l - p ) 2 (23)

1 p pl I <l

where p0 = A/o.

Since Qo is M/D/l it follows that [8]f,2 i _2

Using eqns. 17-19 in eqn. 16 we obtain

c 2 { 1 - P 2 /o< /^ " I l - p 2 , / < / 0

(18)

(19)

(20)

(21)

irnIMMr

signal processing.

Thus care should be exercised when the PDF of interarri-vals has its impulse located at a value greater than theservice time. However, this is not a serious drawback ascondition 23 can be practically satisfied for a wide range oftraffic utilisations. Moreover, although the conditionsderived here are necessary for the noninterference case,they are sufficient for the interfered path case. Forexample, when interfering traffic at the second queue ofFig. 6 exists, p in eqn. 23 increases, and hence the RHS ofthis inequality decreases allowing a wider range of uti-lisations for a given utilisation of the input channel.

H.T. Mouftah received his B.Sc. degree inElectrical Engineering, and the M.Sc.degree in computer science from the Uni-versity of Alexandria, Alexandria, Egypt, in1969 and 1972, respectively; and his Ph.D.in electrical engineering from Laval Uni-versity, Quebec, Canada, in 1975.

From 1969 to 1972 he was an instructorat the University of Alexandria, Researchand Teaching Assistant at Laval Universityfrom 1973 to 1975, Postdoctoral Fellow for

the year 1975-76 at the University of Toronto, and Senior DigitalSystems Engineer and then Chief Engineer at Adaptive Micro-electronics Ltd., Thornhill, Ontario from 1976 to 1977. From1977 to 1979 he worked with the Data System Planning Depart-ment at Bell-Northern Research, Ottawa on several projectsrelated to Computer Communication Networks. In 1979 hejoined the Department of Electrical Engineering, Queen's Uni-versity at Kingston, Ontario, Canada, where he is presently anAssociate Professor. He was consulted for government andindustry in the areas of computer communications and digitalsystems. He holds a number of patents and published a largenumber of technical articles in the area of computer communica-tions, digital systems and multiple-valued logic.

Ahmad Abo-Taleb received his B.Sc. degreein 1977 from Alexandria University, Alex-andria, Egypt and his M.Sc. degree in 1980from Mansoura University, Mansoura,Egypt, both in electrical engineering. Cur-rently he is with the Department of Electri-cal Engineering, Queen's University,Kingston, Canada, completing the require-ments for his Ph.D. His research interestsinclude computer communications net-works, switching techniques and digital

174 1EE PROCEEDINGS, Vol. 132, Pt. E, No. 3, MAY 1985