MULTISERVER QUEUEING NETWORKS AND THE TANDEM QUEUE MODEL

Using a tandem queue model we evaluate the local "endogenous" (- internal) queueing delay in single server and multiserver queueing networks. The new concept of the apparent overall upstream queueing delay(as perceived by the downstream network) allows us to analyze the distribution of this local queue by interpolating between the distributions of the tandem queue (generated by a concentration tree) and the isolated G/G/s queue. The interpolation coefficients depend on the proportion of "premature departures", typically interfering in the upstream stage and leaving the considered path without being offered to the considered local queue. On the other hand, local "exogenous" arrivals (from outside the network) require the introduction of the "interference delay" concept. Finally, in the case of single server queueing networks, we stress the need to extend the capacities o_._f the buffers, by considering the "worst case" scenario and by using an "equivalent tandem queue" model.


Introduction
In this paper, we analyze the local queueing delay of single server and multiserver queueing networks.We assume that customers only gain access to a downstream queue after completion of of the upstream service.Service discipline at all queues is "first come-first servea'.The traditional approximation of networks by means of an isolated G/G/s queue relies too heavily on the use of a local traffic source, as if an output stream from the upstream stage could be considered as a simple traffic source for the downstream stage with no regard to any other influence from the environ- ment.Unfortunately, this modeling approach of the "local traffic source" hides the effects of a number of queueing phenomena which can be significant, especially in the case of single server networks.For instance, models usually ignore the local interference delay caused to an internal ("endogenous") customer by another "ex0genous" cus- tomer arriving directly (from outside the network) at the considered local queue, dur- ing the upstream service time of the preceding endogenous customer.This supplementary queueing phenomenon may be accumulated in the other upstream stages.In addition, the impact of "premature departures" is not completely taken into account in meshed networks.These departures interfere with the upstream progression of cus- tomers, even though these departures are not offered to the considered local queue, since they leave the considered path just before this local queue.Even though the effect of these phenomena is attenuated in multiserver queueing networks, as shown in the results of traffic simulations, this is not a sufficient reason to justify inaccurate models, notably in the case of single server networks.
To take into account the interference between partial traffic streams handled by the considered local queue we consider the concentration tree, over which these traffic streams converge towards that queue.When we introduce the new concept of "apparent" overall upstream queueing delay (as perceived by the local queue), the preceding method of the concentration tree will allow us to consider only the "premature depar- tures" present at the previous stage.It follows that we only need to analyze a simpli- fied network as shown in Figure 1.This can be reduced to the "truncated" network of Figure 2, where partial traffic stream A interferes with "premature departures" of stream B [not offered to final stage (m / 1)] throughout the m-stage tandem queue i (i = 1,...,n).The case of tandem queues of different lengths m will be included in the definition of an overall equivalent tandem queue.To simplify the analysis and to eliminate the influence of the above mentioned interference delay between local "endogenous" arrivals (within the network), we assume that busy periods are not broken up.This assumption is reasonable in the case of heavily loaded networks (when service times do not vary too widely) and is adequate in the case of packet switched networks with the same transmission speed at every stage.It follows (as we will see) that all partial traffic streams have the same distribution of local queueing delay, due to some "agglutination effect" that makes the use of the "interference delay" concept unnecessary.Consequently, this distribu- tion will be evaluated for an arbitrary "endogenous" traffic stream.In other words, the above mentioned case of incoming tandem queues with distinct lengths m can be replaced by the symmetric case of Figure 2 for a given total traffic intensity at the final stage.
In the recent paper [2], we analyzed the case of networks of single server queues showing that the distribution of the local endogenous queue could be derived as an interpolation between the distributions of the isolated G/G/1 queue and the overall tandem queue (equivalent to the above defined concentration tree, jitter delay exclud- ed).It is the tandem queue concept which allows us to make use of the upstream part of the network.The interpolation coefficients depend on the degree to which the network is meshed.The latter determines the proportion of "premature departures".
These coefficients also depend on the heavy load appearing in the considered partial traffic stream.This may explain the impossibility (at the busy hour) of a very high local queueing delay, which is more realistic, compared to the G/G/1 queue, which would lead to a very large queueing delay!In the present work we extend this analy- sis to the case of multiserver networks and show to what extent the approximation by the isolated G/G/s queue is more easily justified.
In Section 2 we outline the earlier study of single server queueing networks before extending the analysis in Section 3 to the case of multiserver networks.Finally, in Section 4 we conduct a numerical study and compare the values obtained with the re- sults of traffic simulations in the case of packet switched networks with two links arbi- trarily handling two populations of packets of very different lengths leading therefore to widely differing service times.In Section 2 and 3 the arrival processes at the net- work input are assumed to be governed by a general probability distribution in the stationary regime.In Section 4 the study is limited to the case of Poisson arrivals and we evaluate the traffic handled in the buffers, particularly in the case of single links.In the latter case, we stress the need to extend the capacities of the buffers by using an "equivalent tandem queue" model.
2. Single Server Queueing Networks 2.1 Notation and Assumptions Let us consider the truncated network of Figure 2. We recall that the queueing discipline (in each successive queue) is "first come-first served".The system is assumed to be in the sationary regime.

Stage
Stage m Stage (m+ For each of the n identical and independent tandem queues of Figure 2, at stage k (k-1,..., m) and for the jth customer at the considered queue, we set: arrival epoch at stage k" Xk; interarrival time between customers (j-1)and j" yk_ 1-=X k --X k-1; occasional idle period during the interarrival interval yk_ l'e k" Finally, for the considered final stage server, the sequence of arrivals will be indexed by i with the processes: interarrival interval at the input to all n tandem queues, for the customers offered to the considered final server: Yi-1; Note the difference between the arrival process Y-I for a given tandem queue and the process Yi-1 relating to all tandem queues but only for customers destined for the considered final stage server.The couple [Yi_l,Ti] defines an isolated G/G/1 server handling the same partial traffic streams as those handled by the con- sidered final stage server.Its queueing delay will be denoted W o in the stationary re- gime.In this regime, the arrival processes Y-I and Yi-1 are governed by some general probability distribution.

The Jitter Effect
To be able to use the concept of equivalent tandem queue in order to evaluate the local queueing delay at the final stage, we need to replace the actual process Y_ 1 by a process keeping the same arrival order as at the entry to the network, since a tandem queue keeps the same arrival order at each stage.The difference between the two processes comes from the mutual independence of the n incoming tandem queues in Figure 2.
This difference generates a local jitter effect.All the customers of the same local busy period experience the same local jitter delay J, whose distribution function has been approximately evaluated in Le Gall [2], Subsection 3.1, for large n (Figure 2): p,,t T with T" T p,, 1 p" 1 p. 1 T This jitter effect is only significant for heavily loaded networks. (1)

The Equivalent Tandem Queue
Firstly, we assume that there are no "premature departures" (i.e., no traffic streams B in Figure 2) and no local "exogenous" (coming from outside) arrivals.The local queue at the final stage of the concentration tree (already mentioned in the introduction) may then be equivalent to that of an equivalent tandem queue as defined in Le Gall [2], Subsections 2.1, 2.5 and 2.7.To understand this, consider the relations (at stage k-1 < m): (5) Tj +wj 8j_ 1 If the busy periods are not broken up, the downstream busy period corresponds to the upstream busy period.Consequently, during the downstream busy period (at stage k) we have no idle period, e k-1 0, and k-1 k k (6) T j + wj sj l.
The arrival process disappears during the local busy period.Its influence only appears to initiate the busy period through the jitter effect.Finally, we do not change the local queueing delay (excluding the jitter effect) if we replace a nonsymme- trical network by the symmetrical network of Figure 1, provided we keep the same total traffic intensity and the same length for the equivalent tandem queue.
From (6), we deduce that the busy period is not broken up if the following condi- tion is satisfied: In the case of packet switched networks, we have Tj T j, if the transmission speed is the same at each stage.Relation (6) becomes: k k =s k.
Sj--Sj_ 1 =... 30 (8)   where Jo corresponds to a customer initiating the busy period.The sojourn time has the same value for all the customers of the same busy period: it corresponds to some agglutination phenomenon, which also appears at the final stage of the equivalent tandem queue.
Moreover, relation ( 7) may be satisfied even in the case of mutually independent successive service times for heavily loaded networks when the extended delays at the final stage tend to cause busy periods to amalgamate, if service times do not vary too widely.(See Hypothesis 2 in Section 2.7 of [2].) 2.4 The Impat of Premature Departures Now we suppose that "premature departures" take place at the upstream stage.
They are not offered to the final stage server considered and correspond to traffic streams B in Figure 2.
To evaluate the local queueing delay we have to subtract the overall upstream queueing delay V(1;m), from stage 1 to stage m, from the overall queueing delay W(1; m + 1) from stage 1 to stage (m + 1). in fact, in Le Gall [2], we showed that we do not have to use the observed value V(1;m) related to all the upstream customers.
Observe that the upstream customers considered are offered to the final stage server considered, with a probability (l/n) in Figure 2. Finally, we use the "apparent" over all queueing delay as perceived by the local queue in the form V'( h V(1; where h is the random variable 1 with probability (l/n) and 0 with probability [1-(i/n)].We will also introduce this "apparent" delay in the evaluation of W(1; m / 1), since we wish to evaluate the local queueing delay.
In the general, nonsymmetrical case, we define at upstream stage rn (including all the incoming tandem queues)" total load traffic intensity): a; part of this total load corresponding to the customers offered to the considered final stage server: a'; part of this total load corresponding to "premature departures": a".We have: a a'+ a".
(10) Following our comment after relation (6), we may replace this nonsymmetrical case by the symmetrical case of Figure 2 where the number of identical incoming tandem queues is: n integer part of a/a'.
(11) The length m of these identical tandem queues was defined in Le Gall [2], Section 2.7.Now, in relation (9) the random variable h is defined by expression (11).

The Local Queueing Delay Without Local Exogenous Arrivals
We suppose that all local customers come from an upstream stage ("endogenous arrivals").No customer arrives directly from outside the network ("exogenous arrivals").In Le Gall [2], Theorem 5, we derived the key formula for the local queue- ing delay w of an arbitrary customer of any partial traffic stream, at the final stage server considered in the steady state: (12) where D(m) is the local queueing delay of the "concentration tree", i.e., of the (m + 1)-stage equivalent tandem queue including the jilter delay at the final stage.
Finally, expression (12) combines the case of no premature departures or heavy load in the considered partial traffic stream [local queueing delay D(m) with actual upstream load] and the case of many premature departures corresponding to a local queueing delay W o of the isolated G/G/1 server.This simple result is due to the fact that the busy periods are not broken up.Moreover, it explains why a heavy endoqenous load cannot generate a very large local queueing delay (as opposed to an isolated G/G/1 queue).

Case of Local Exogenous Arrivals
We suppose the existence of local exogenous arrivals at the final stage server (i.e.not coming from the upstream stage) with the following notation in the stationary mode: total load traffic intensity): p; endogenous load: Po; exogenous load: Pl" We have: p=po+Pl (13) We include the existence of exogenous arrivals for the evaluation of W o and D(m) in expression (12) by supposing that the exogenous arrivals are offered to stage 1 with zero service time in successive stages 1 to m and with the actual service time at the final stage (m + 1).But to apply relation (9) we have to take care, as we will explain below.
In the evaluation of W o and D(m), we suppose that the interferences are only due to the arrivals (at the entry to the network) and to the service times.Now, we note that the exogenous arrivals may be immediately served during the overall upstream queueing delay of some endogenous arrivals generating a new interference delay.To take this phenomenon into account in relation (9) we will distinguish between the two kinds of local arrivals.
For an arbitrary endoqenous arrival, we note that an upstream customer may only be offered to the considered final stage server if this server is not busy due to an exogenous arrival (probability of occupancy: Pl)" Finally, in expression (12) we have to make the substitution: a' a' a '-if" (1 Pl)" ( 14) For an arbitrary local exoqenous arrival, expression (9) says that the considered final stage server is busy due to an arbitrary endogenous arrival.In expression (12)   we have to make the substitution: a' a' -a--a P0" (15)  This method of evaluation could not be a part of the classical concept of local traffic source.

Multiserver Queueing Networks
Now, when considering the networks of Figure 1 and 2, we replace each single server by a multiserver with a capacity of L single servers.We retain the same load for each new server as in the previous case and we keep the same service discipline (FC-FS) for each queue.Finally the arrival process yl_ 1 and Yi-1, as considered at the last paragraph of Subsection 2.1, are still governed by some general probability distri- bution.We want to extend expression (12), which needs some preliminary remarks to use the concept of the equivalent tandem queue.

The Equivalent Tandem Queue
Firstly, we need to figure out to what extent can we go when using relations similar to (2)-( 7) above.Let us consider a G/G/L queue with the vectorial equation given by Borovkov [1].At stage k < m), the interarrival time is still denoted Y/ 1" Let wk.h be the queueing delay uil (at least) h servers are free for arrival X-The 3, J" vector w k .J has coordinates w k.3,h(h -1, ,L).Let R(x) be the vector with entries (Xl,...,XL) where x 1 Minh(Xh) and e (1, O, ., O) and i= (1,...,1).Analogous to relations (2), we may write at stage (k-1): A Minh(w j 1,h) + Tj ek-1 is positive on an idle period or due to a partial occupancy at stage (k-1) during the time Yk_-ll.Consequently, we have:

Tj +ej
Let us apply this expression to (16).We deduce the following relation, similar to k e)i ek-l] k 1.i + k na T l"i,R(W i+Tj_I Tj wj (19)   As for the single server case, we will suppose that the busy periods are not broken up and, consequently, e-1 0 during the local busy period, leading to: Tjk-l i-f-wjk l (w -1 zr Tj_ e).This expression is consistent with condition (7) to avoid breaking up the busy periods and, consequently, the same customer initiates the busy period in each success- ive upstream stage.The same arguments as for the single sever case may be used to introduce the concept of equivalent tandem queues if there are no "premature depar- tures".But there are two apparent differences: a multiserver tandem queue cannot keep the same arrival order at each successive stage: consequently, the local jitter effect disappears; unless successive service times are constant, we cannot define the string of successive servers used by the same customer (particularly in the case of interferences with "premature departures") and, consequently, we cannot evaluate with accuracy the number (m + 1) of stages for the single server equivalent tandem queue.Due to the slight impact of m, we will approxi- mate the equivalent tandem queue by taking m-1.
On the other hand, from ( 22), we may deduce again that the local queueing delay (at the final stage) does not change if we replace a nonsymmetrical network of multiserv- ers by a two-stage symmetrical network of single servers as illustrated in Figure 2.
The local queueing delay at the final stage of the equivalent two-staqe tandem queue will be denoted D(1).

The Local Queueing Delay Without Local Exogenous Arrivals
Provisionally, we assume no local exogenous arrivals.But now, "premature depar- tures" exist, as illustrated in Figure 2. At the final stage, a customer, served just after another customer from the same incoming single server tandem queue (see above), perceives only a final stage single server due to the property (22) of the local busy period.On the contrary, he perceives a final stage multiserver if he is disturbed by upstream "premature departures".We introduce a slight modification in the no- tation of Section 2.4, at the upstream stage (for the total number of servers): a: total load of the network; a': part of this total load corresponding to the customers offered to the con- sidered final stage multiserve (a']L): part of this total load handled by the considered equivalent single server tandem queue.
In the equivalent, symmetrical case, the number n of incoming single server tandem queues is, instead of (11): n-integer part of (a'/L)" Relation (9) has to be applied with this new value of n and, instead of (12), the local queueing delay w for an arbitrary customer of any partial traffic stream, at the final stage multiserver in the stationary regime, is defined by" where D(1) is the local queueing delay at the final stage of the equivalent two-stage single server tandem queue (m-1, without any jitter effect), and W o is the local queueing delay of the isolated G/G/L server.But now, the interpolation coefficient is.L times smaller than in the case L-1.In practice, with more than three incom- ing paths of multiservers (and L > 2), we have approximately: Ee" " Ee " W. ( In that case, the result is consistent with the concept of a local traffic source.We may mention that the explicit distribution of W o has been recently given in Le Gall Note: In the case of constant service times, D(1)--0 in ( 24), but the jitter effect appears again.

Case of Local Exogenous Arrivals
Now, we suppose the existence of local exogenous arrivals at the final stage multi- server.The notation of Section 2.6 is still valid for each server of the considered final stage multiserver: case of the loads P0 and Pl per server.The concept of equivalent single server tandem queue allows us to substitute ( 14) and (15) in expression (24) to define the local queueing delay for an endogenous arrival and for an exogenous arri- val, respectively, provided that D(1) and W o include the exogenous arrivals.

Case of Multilink Packet Switched Networks
We will apply the preceding considerations to the case of multilink packet switched networks with Poisson arrivals.

The Traffic Model (L >_ 1)
We consider the symmetrical network with n branches of m successive multilinks (-multiservers), of capacity L, and a final stage multilink (of capacity L) as illus- trated in Figure 2. The system load p is the same in each link of the multilinks in successive stages.The arrival rate (in each link) for each individual traffic stream A and B is A. The transmission speed is the same at each stage, i.e., the successive packet lengths (-service times) for the same customer are identical: Tin-T2n-...= Tn m Tn, with T-E(Tn).
Each individual traffic stream (in each link) is the mixture of two partial traffic streams of category j (j-1,2), corresponding to packets of constant length T:i (T < T2) Aj being the arrival rate.We let: pj-Aj.Tj, A-A + A2, /9-Pl -I-P2, T-p/A.
For (case of an arbitrary customer), Tables la and l__b_b give comparative results between simulations and calculations [formulae ( 24) and ( 27)] for p-0.9 (Table la) and p-0.6 (Table lb) with two cases: P2-Pl and P2-0"3P1" For n-10 there does not appear to be any significant discrepancy between and W o. For n-3 and p-0.6, on the contrary, is 30% (for P2-Pl) and 40% (for P2-0"3pl) higher than W o (M/G/2 queue).But even in this case, there appears to be a good agreement between simulations and calculations [formula ( 24
short packets

Lq packets
Total packet length: T 1 1 T 2 10 arrival rate (per link): A 1
4.3 The Occupancy in the Buffers a) Case o th..._e multilink networks (L > 1) We use the traffic model as described in Subsection 4.1 for the local queue consider- ed, but we drop the hypothesis: (a/a')-n.Expressions ( 24) and ( 27) give for the mean local sojourn time of an arbitrary packet: a --a'-D(1) + (1 -)Wo + T, wit : -K In expression (27) of D(1) we may use the approximation: -e 1 We g rewrite: 1 P. (T 2 T1), with: H-1-p A 1-P1" gives the occupancy in the buffer per arbitrary packet, when the environment is at busy hour.When the environment is a off-peak hour (i.e.very small traffic inten- sity), we have: a'a and expression (29) becomes closer to the tandem queue model" Practically, for dimensioning purpose, we have to consider this "worst case".At the same time for the local queue considered in the network, the occupancy (per arbitrary packet) is the smaller value T, given by expression (28).
b) Case o_ sinqle link networks (L 1) In the case of single link queueing networks (L 1), we deduce that the buffers have to be overdimensioned, on using the "equivalent tandem queue" model and not on using the traditional G/G/1 model.Due to expression (29) of D(1), a proportion [p2/(1-pl)] of short packets has the same high occupancy in the buffer as long packets, when the environment is not at busy hour.It is due to the aqqlutination phenomenon of short packets behind long packets as already mentioned after relations (8).Practically, the "worst case" corresponds to off-peak hours in the network, except for some traffic streams where each incoming path corresponds to only one outgoing path, this path handling the same traffic stream and only this traffic stream.For this "worst case", the capacity of buffers has to be multiplied by 3 (and even by 4) compared with the use of the traditional G/G/1 model, when T 2_> 30.T 1.
and" K(rn) m. 1 fll For n-1, the jitter delay does not exist: J-0.With the traditional M/G/1 server model, the buffer is dimensioned for an occupancy equal to (W 0 + T).With the tandem queue model the occupancy becomes D(m)+ T. It follows an overload coefficient (of the buffer) equal to: C= D(m)+T. (32)
(33) As a consequence, it is absolutely necessary t._qo dimension th...Ae buffer on usinq the tan- dam.queue model, because the buffer may be congested before any detection by the time-outs in the network: in our example p-0.6 only!And during all this time con- gestion of the buffer, all the links under the control of this buffer are not accessible.For example, as we mentioned above, it is the case of the occurrence of some occasion- al overloads in some traffic streams at off-peak hours in the network.Finally, the phenomenon is significant because the busy hour in the buffers is quite different o_o_f th..__e busy hour in the network.

Conclusion
Based on the method in [2], we used the tandem queue model.We exploited the con- cept of apparent upstream queueing delay as perceived by the downstream network, to be able to introduce simply an interpolation linking the distribution of the local

Figure 1 :
Figure 1: The full network