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1 Computers & Operations Research 37 (2) 74 8 Contents lists available at ScienceDirect Computers & Operations Research ournal homepage: On tem blocking queues with a common retrial queue K. Avrachenkov a,, U. Yechiali b a INRIA Sophia Antipolis, France b Tel Aviv University, Israel article info Available online 7 October 29 Keywords: Retrial networks Mean value analysis Fixed point approach abstract We consider systems of tem blocking queues having a common retrial queue. The model represents dynamics of short TCP transfers in the Internet. Analytical results are available only for a specific example with two queues in tem. We propose approximation procedures involving simple analytic expressions, based on mean value analysis (MVA) on fixed point approach (FPA). The mean soourn time of a ob in the system the mean number of visits to the orbit queue are estimated by the MVA which needs as an input the fractions of blocked obs in the primary queues. The fractions of blocked obs are estimated by FPA. Using a benchmark example of the system with two primary queues, we conclude that the approximation works well in the light traffic regime. We note that our approach becomes exact if the blocking probabilities are fixed. Finally, we consider two optimization problems regarding minimizing mean total soourn time of a ob in the system: (i) finding the best order of queues (ii) allocating a given capacity among the primary queues. & 29 Elsevier Ltd. All rights reserved.. Introduction Maority of TCP transfers in the Internet are small in volume, consisting of only few packets [6]. The TCP congestion control mechanism does not have a chance to influence the dynamics of the traffic originated from short TCP transfers. Many short TCP transfers fit in the minimal size congestion window hence the rate of the TCP transfer cannot be controlled by means of congestion window. We argue that for such type of TCP traffic, a network of blocking queues with retrials is an appropriate model. Then, an additional motivation for the study of retrial networks with blocking finite buffer capacity queues is the drop tail queue management policy employed in the Internet routers. A router using drop tail policy drops packets from the end of the queue when the queue size increases beyond some value. The dropped packets are then retransmitted by the sender. Explicit analytic results were derived in [4] for a system comprised of a single M=M== primary (blocking) queue an associated M=M== retrial (orbit) queue from which blocked obs from the primary queue retry to be processed. Further explicit results were obtained for a system with two M=M== queues in tem a common associated M=M== orbit queue. The case with two queues in tem turned out to be involved enough to predict that exact analytic solutions for r 42 tem queues with blocking common associated retrial queue will be very difficult to achieve, even if achieved, the expressions for the various performance measures will be extremely complicated hence with no significant insight. Therefore, in this work, we propose approximation procedure consisting of two parts. In one we use mean value analysis (MVA) to derive simple analytic expressions for the mean number of visits to the orbit queue the mean soourn time of a ob in the system. The obtained expressions use as parameters the fractions of blocked obs. Thus, in the other part of our approximation procedure we estimate the fraction of blocked obs with the help of a fixed point approach (FPA). By comparing the approximation results with the exact results for the case of r ¼ 2 queues, we show that the proposed approximation is good when the system load is light. Specifically, in the mean value analysis, assuming a fixed probability p of blocking in queue, we calculate the probability generating function (PGF) mean of N, the number of times an arbitrary ob visits the orbit queue before passing queue (rrr) for the first time, where N r specifies the total number of times an arbitrary ob visits the retrial queue before leaving the system. We then derive the Laplace Stieltes transform (LST) calculate the mean of Y, the total soourn time of an arbitrary ob in the system until it passes queue for the first time. Similarly to N r, Y r specifies the total soourn time of a ob in the system. In the fixed point approach we assume that the input flows are Poissonian we use Erlang s loss formula for the M=M=c=c queue. Having these results we consider two optimization problems: Corresponding author. Tel.: addresses: k.avrachenkov@sophia.inria.fr (K. Avrachenkov), uriy@post.tau.ac.il (U. Yechiali). (i) Finding the best order of arranging the queues so as to minimize the mean total soourn time of a customer in the /$ - see front matter & 29 Elsevier Ltd. All rights reserved. doi:.6/.cor.29..4

2 K. Avrachenkov, U. Yechiali / Computers & Operations Research 37 (2) system, when the orbit queue is either an M=M== system or an M=M== system. We show that the optimal order is to arrange the queues in an increasing order of the index ð p ÞE½B Š=p, where B is the processing time of a ob in primary queue. (ii) Given a fixed total capacity C to all r queues, how this amount of resource should be allocated to the various queues so as to minimize the total soourn time of a ob through the system. In comparison with the single node retrial queues [,3,7,8], the networks of queues with retrials receive significantly less attention. In [2] the authors prove the non-existence of productform solutions for certain queueing networks with retrials. Jackson-type systems with r tem non-blocking M=M== queues with feedback to (i) the first queue, (ii) to a common M=M== retrial queue, where feedback from each queue to the retrial queue is applied only after a ob passes queue, have been analysed in [5]. The following related model was also studied in [9]: a single ob is made up of r independent tasks, all of which must be successfully performed for the ob to be completed. Upon failure at any stage, the ob has to be started all over again. 2. The model Consider a system with r blocking primary queues in tem, a common associated retrial (orbit) queue to which all blocked obs from the various primary queues are dispatched. Each blocked ob, after spending a soourn time in the orbit queue, tries to be admitted to the first queue then continue traversing successfully through all r queues, until finally leaving the system. Thus, a ob may traverse mor queues only to be blocked in the (mþ)-th queue, then, after spending time in the orbit queue, start all over again from the first queue. A schematic presentation of the system is depicted in Fig.. Assume that the outside arrival rate of new obs to the system is l obs per unit time. Assume for a while that the blocking probabilities P ( ¼ ; 2;...; r) in the various primary queues are fixed. That is, P ¼ p. (Further assumptions will be introduced for the various scenarios treated in the ensuing sections.) We first calculate the probability generating function (PGF) mean of the number of times a ob visits the orbit queue until leaving the system. We then derive the Laplace Stieltes transform (LST) mean of the time it takes to achieve that. 3. Number of visits at the orbit queue Let N be the number of times a ob visits the orbit queue until it passes successfully queue for the first time. For Z wehave (N ¼ ) ( N ¼ N w:p: p ; N þþn w:p: p ; where N is an independent replica of N. We thus have that N ðzþ, the PGF of N, is given by N ðzþ¼e½zn Š¼ ð p ÞN ðzþ zp N ðzþ ; E½N Š¼ E½N Šþp p : Iterating with N ðzþ¼ð p Þ=ð zp Þ with E½N Š¼p =ð p Þ we get that Q N ðzþ¼ i ¼ ð p iþ zð Q i ¼ ð p iþþ ; E½N Š¼ Q i ¼ ð p iþ Q i ¼ ð p ¼ X iþ p m Q i ¼ m ð p iþ : It follows that N has a geometric distribution (shifted to ) with success probability Q i ¼ ð p iþ. Clearly, as mentioned, N r is the total number of times a ob visits the orbit queue until it successfully leaves the system. It follows that with fixed blocking probabilities, the total number of times a ob visits the orbit queue, until successfully passing queue, isindependent of the order of any set of primary queues, for every rrr. Indeed, a ob passes queue if only if it is not blocked in any of the first queues, which occurs with probability Q i ¼ ð p iþ. This explains why N is independent of the order of those queues. Remark. For the calculation of N ðzþ E½N Š when the blocking probabilities are fixed, the primary queues can be of any blocking type they need not be all the same. 4. Soourn time of a ob in the system Let the service time of a ob in queue be a rom variable, B ð ¼ ; 2;...; rþ, having a general probability distribution function. The soourn time of a ob in queue is denoted by W. Assume further that each time a ob visits the orbit queue it resides there for a rom time, W. Naturally, this rom time depends on the assumptions on the type of queue the orbit queue is (e.g. G=G==, M=G==, or M=G==, etc.). Thus, if for example the orbit queue is an =G==, where the service time is B, then W ¼ B. Let Y be the length of time until a ob first passes successfully primary queue. Then, similarly to the derivation of N, we can write (Y ¼ ) ( Y ¼ Y þw w:p: p ; Y þw þy w:p: p ; where Y is an independent replica of Y. Orbit Queue Queue Queue 2 Queue r Fig.. Scheme of the system.

3 76 K. Avrachenkov, U. Yechiali / Computers & Operations Research 37 (2) 74 8 Thus, the LST of Y, Y ðsþ¼e½expð sy ÞŠ, is given by Y ðsþ¼ ð p ÞY ðsþw ðsþ p Y ðsþw ðsþ ; its mean by E½Y Š¼ E½Y Š þ p Šþ p p Š: Iterating with E½Y Š¼½p =ð p ÞŠ Šþ Š, we obtain E½Y Š¼ X ¼ X m Š Q i ¼ m þ ð p iþ þ Š X m Š Q i ¼ m þ ð p iþ þe½n Š Š: p m Q i ¼ m ð p iþ Now, the mean soourn time of a ob in the system is given by E½Y r Š. 5. Minimizing the mean soourn time (when blocking probabilities are fixed) Our obective now is to arrange the queues so that E½Y r Š, the mean total soourn time of a ob in the whole retrial network, is minimized. Since E½N r Š is independent of the order of the queues, it suffices (see ()) to find the order of queues that minimizes X r m Š i ¼ m þ ð p iþ : Let p ¼ð; 2;...; ; ; þ; þ2;...; rþ be the order (policy) that arranges the queues according to some initial order ð; 2;...; rþ. Let p ¼ð; 2;...; ; þ; ; þ2;...; rþ be the policy in which the order of queues þ is interchanged with respect to p. Set a m Š m ¼ i ¼ m þ ð p iþ : Then, under p,wehave E½Y r p Š¼ X while, under p,wehave a Š m þ i ¼ þ ð p iþ þ þ Š i ¼ þ 2 ð p iþ þ E½Y r p Š¼ X a þ Š m þ ð p Þ i ¼ þ 2 ð p iþ Š þ i ¼ þ 2 ð p iþ þ Xr a m : m ¼ þ 2 Xr m ¼ þ 2 Thus, after multiplying throughout by i ¼ þ 2 ð p iþ, it follows that E½Y r p ŠrE½Y r p Š if only if Š þ p þ Šr þ Š þ þ p Š: That is, p is better than p if only if p p Šr p þ p þ Š: ð2þ þ By repeating queue interchanges we conclude that E½Y r Š is minimized if only if the queues are arranged in an increasing order of the index p p Š: That is, if p is large, then the mean number of attempts until first passing queue, namely p =ð p Þ, is also large, hence it is ðþ a m ; better to place queue at the beginning of the network of tem queues. Similarly, small Š has the same effect. Remark 2. If each of the primary queues is a =G== queue with B being the service time of a ob, ð p Þ being the admission probability, independent of the state of the system, then W ¼ B for every rrr the optimizing index is p p E½B Š: 6. Fixed point approach Let l be the external arrival rate to primary queue. We first calculate the overall input rate to each primary queue, as well as to the orbit queue. Let L denote the overall input rate ( ¼ mean number of arrivals per unit of time) at the gate of primary queue. If the blocking probability at queue is P (P can be interpreted as the long time average fraction of obs sent from queue to the orbit queue), the arrival rate to queue r must be L r ¼ l=ð P r Þ, since L r ð P r Þ¼l obs enter leave the stationary system per unit of time. The blocked rate L r P r is directed to the orbit queue. Similarly, L r ¼ L r =ð P r Þ L ¼ L þ =ð P Þ for rrr. This implies that L ¼ l= i ¼ ð P iþ. Thus, the overall rate of blocked obs arriving at leaving the orbit queue is L ¼ Xr ¼ L P ¼ l Xr ¼ P i ¼ ð P iþ ¼ le½n rš: Indeed, since E½N r Š is the mean number of times a ob visits the orbit queue, the output rate of that queue is L ¼ le½n r Š. Now, clearly, L ¼ lþl ¼ l@ þ Xr ¼ P i ¼ ð P A l ¼ iþ i ¼ ð P iþ : Suppose now that each primary queue is a =G=K =K queue. Assume further that the arrival rate to each queue is approximately Poisson, implying that each primary queue is an M=G=K =K queue with arrival rate L. Then, the blocking probability P of queue can be approximated by the Erlang loss formula, namely, ~P ¼ rk =K! P K i ¼ ri ð3þ ð4þ =i! ; ¼ ; 2;...; r; ð5þ where the approximated offered load at queue is calculated as r ¼ L E½B Š¼ le½b Š i ¼ ð P ~ i Þ ¼ L þ E½B Š P ~ : Thus, for queue r, r r ¼ L r E½B r Š¼ le½b P Kr rš P ~ i ¼ ¼ le½b r Š ri r P =i! Kr r i ¼ ri r =i! ¼ le½b rš þ rkr r =K r! P Kr i ¼ ri r =i! The above equation determines the value of r r, from which ~ P r is readily calculated. Now, we can write L r ¼ L r ~ P r ; r r ¼ L r E½B r Š ¼ le½b r Š ð ~ P r Þð ~ P r Þ ¼ le½b r Š ð ~ P r Þ þ rk r r =K r! P Kr i ¼ r i r =i! Then, going down from r to, all r can be calculated along with all ~ P.! :! :

4 K. Avrachenkov, U. Yechiali / Computers & Operations Research 37 (2) To check the validity of this fixed point approach we will compare, for each, the above probability ~ P with the fraction of times P a ob is blocked at queue. 7. Calculating the load-dependent blocking probabilities for a network with M=G == primary queues Suppose (see Section 6) that each queue is an M=G== type queue. That is, we make the approximation that the arrival flow to queue, at a rate of L ¼ l=ð m ¼ ð P mþþ is Poissonian. This assumption implies that the mean interarrival time to queue is =L ¼ð m ¼ ð P mþþ=l. Hence, the long run average blocking probability in queue (being an M=G== queue, or using Erlang s loss formula with K ¼ ) is ~P ¼ b ð m ¼ ð ~ P m ÞÞ=lþb ¼ lb m ¼ ð ~ P m Þþlb ¼ g m ¼ ð ~ P m Þþg ; where b :¼ E½B Š g :¼ lb for ¼ ; 2;...; r. Under p we have g ~P r ¼ r ð P ~ : ð7þ r Þþg r Eq. (7) is a quadratic equation in P ~ r its solution is P ~ r ¼ g r (the solution P ~ r ¼ is not of interest). Indeed, since every ob enters queue r once only once, the load on this queue is g r ¼ lb r this is the fraction of time queue r is busy hence, it is also its blocking probability. It follows that L r ¼ l=ð P ~ r Þ¼l=ð g r Þ. Now, for queue ¼ r, the inter-arrival time is =L r ¼½ m ¼ r ð P ~ m ÞŠ=l. This implies, using (7), that g ~P r ¼ r g ð P ~ r Þð P ~ ¼ r r Þþg r ð P ~ : ð8þ r Þð g r Þþg r The solution of the quadratic equation (8) is ~ P r ¼ g r =ð g r Þ. We therefore claim. Lemma. The blocking probabilities are given by g ~P ¼ ; ¼ r; r ;...; 2; ; ð9þ s þ where s ¼ P rm ¼ g m (s r þ ¼ ). Proof. The lemma has been shown to be true for ¼ r r. We assume that it holds for all ¼ r; r ;...; kþ prove its validity for ¼ k. We first claim that m ¼ k þ ð P ~ m Þ¼ s k þ. This follows by substituting from (9) the values of P ~, ¼ r; r ;...; kþ. Thus, g ~P k ¼ k g m ¼ k ð P ~ ¼ k m Þþg k ð P ~ : ðþ k Þð s k þ Þþg k Again, the solution of () is P ~ k ¼ g k =ð s k þ Þ, which completes the proof by induction. & We note that from (9) it follows that P ~ k o if only if s k ¼ P k ¼ lb o. Indeed, it has been shown in [4] that for a retrial tem network with two M=M== primary queues, where m ¼ m ¼ =b ¼ m 2 ¼ =b 2, a necessary condition for stability is m42l. That is 42l=m ¼ lb þlb 2 ¼ s 2. Moreover, when the retrial queue is a =M== queue with mean service time b ¼ =m, it has been shown in [4] that when m -, a necessary sufficient condition for stability becomes again s 2 o. 8. Capacity allocation Assume that the total capacity budgeted to the primary nodes of the tem network is m, that is, P r ¼ m ¼ m. We would like to ð6þ distribute the total capacity in some optimal way among the primary queues. We consider separately two case. 8.. Blocking probabilities are fixed If P ¼ p, independent of the queue load, then the optimization problem is (when m Š¼b =m m ) ( ) min E½Y r Š¼ Xr =m m Qr i ¼ m þ ð p iþ þe½n rš Š subect to X r m m ¼ m; m m 4; ; 2;...; r: ðþ With E½N r Š independent of the m s, by using Lagrange multipliers differentiation one gets that the optimal values of m s satisfy m 2 þ ¼ð p þ Þm 2 ¼ Y þ i ¼ 2 Thus, we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ Xr u m þ t ð p i ÞA m ¼ 2 i ¼ 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Y m t ð p i Þ i ¼ 2 Am ð p i Þm 2 ; rrr : m; ð2þ ; 2rrr: ð3þ That is, in the optimal capacity allocation, the first queue gets the largest capacity then each following queue gets a smaller pffiffiffiffiffiffiffiffiffiffiffiffiffi capacity, reduced by a factor of p Blocking probabilities estimated by P ¼ g =ð s þ Þ In the case when the blocking probabilities are estimated by P ¼ g =ð s þ Þ, then E½N r Š (Section 3) does play a role. We use P ¼ð s Þ=ð s þ Þ i ¼ m ð P iþ¼ s m. Thus, the optimization problem becomes: ( min E½Y r Š¼ Xr =m m ð s m þ Þ þ Š X ) g m =ð s m þ Þ r ð s m Þ subect to X r m m ¼ m; m m 4; ; 2;...; r: ð4þ Recall that s m ¼ P r i ¼ m lb i ¼ P r l=m i ¼ m i g m ¼ l=m m. Using Lagrange multipliers for problem (4) does not yield a nice solution, but it can readily be solved numerically by stard procedures. As we have noted above, the term with E½N r Š cannot be neglected in this case. However, when E½N r Š is small (e.g., when the retrial queue is =M== queue m is large), we can apply the results of Section 5. In particular, in Section 5 it was shown that E½Y r Š is minimized if the index P ~ b ¼ s b ¼ ~P l ð s Þ g is increasing. However, ð s Þ is increasing for any order of the queues. That is, all orders give the same mean total soourn time. This result seems at first to be somewhat surprising. However, numerical calculations performed in [4] for an analytic, nonapproximating, solution of a network of two (r ¼ 2) M=M== type queues (with common M=M== retrial queue) showed that L system, the mean overall number of obs in the system is symmetric with respect to the mean service rates m m 2 for a given value

5 78 K. Avrachenkov, U. Yechiali / Computers & Operations Research 37 (2) 74 8 of m þm 2. That is, any order of the two queues will result in the same value of L system. 9. Numerical results Here we perform numerical comparison of proposed approximations versus Monte Carlo simulations exact results available for a particular case. Specifically, in [4] we explicitly solved the model with two (r ¼ 2) M=M== tem queues an M=M== orbit queue. We shall refer to the results of [4] as the exact model. Let us recall some results from [4]. The mean total soourn time of a ob in the system T system is, using Little s law, T syste l L system; where L system denotes the average number of obs in the system, given by (see Eq. (3) in [4]) L system ¼ L orbit þp ðþþp ðþþ2p ðþ; where P i ðþ is the probability of i obs in queue obs in queue 2 (i; ¼ ; ). The probabilities P ðþ, P ðþ P ðþ, representing the fraction of time the system is in state (,), (,) or (,), respectively, were found to be (see Proposition 3 in [4]) P ðþ ¼ l m ; P ðþ ¼ lðm m 2 ðm þm 2 þm Þ lðm ðm þm 2 Þ m m 2 Þ l 2 ðm þm 2 ÞÞ m m 2 2 ð2lþm þm 2 þm Þ ; P ðþ ¼ l m 2 P ðþ; while L orbit was shown to be L orbit ¼ L þl þl þl ; with L, L, L L being calculated from the set of linear equations (26) (29) in [4]: ðlþm ÞL m 2 L ¼ ; ðlþm ÞL m L þm 2 L ¼ lp ðþ ðl m ÞP ðþ m 2 P ðþ; m L þðlþm 2 þm ÞL m L ¼ m P ðþ; m L ll þm L ðlþm ÞL ¼ lp ðþþðlþm ÞP ðþ: Let us compare T system E½Y 2 Š, where E½Y 2 Š¼ 2 Šþ Š P ~ þe½n 2 Š Š; ð5þ 2 with! Š¼E½B Š; ¼ ; 2; E½N 2 Š¼ P ~ P ~ 2 P ~ þ P ~ 2 : To estimate Š we assume the orbit queue to be of an M=M== type with arrival rate L mean service time E½B Š¼=m. Thus, Š is given by Š¼ ¼ m L m le½n 2 Š : For the 2-queue in tem M=M== orbit queue from [4] we can calculate the exact long time average fraction of obs blocked at each primary queue. Namely, the blocking rate at the gate of the first primary queue is L P ¼ lðp ðþþp ðþþþm ððp ðþ P ðþþþðp ðþ P ðþþþ; where P i ðnþ is the probability of i obs in queue, obs in queue 2 n obs in the orbit queue, L ¼ lþl ; L ¼ m ð ðp ðþþp ðþþp ðþþp ðþþþ; is the rate of obs coming out of the orbit queue, while P ðþ, P ðþ, P ðþ P ðþ are given in Proposition 3 of [4]. Thus, we have P ¼ lðp ðþþp ðþþþm ððp ðþ P ðþþþðp ðþ P ðþþþ : ð6þ lþm ð ðp ðþþp ðþþp ðþþp ðþþþ The rate L 2 is given by L 2 ¼ m ðp ðþþp ðþþ; the rate of blocking at the gate of the second primary queue is L 2 P 2 ¼ m P ðþ: Thus, we can write P ðþ P 2 ¼ P ðþþp ðþ ¼ l=m 2 P ðþ l=m þl=m 2 P ðþ : ð7þ Specifically, ¼ þ P ðþ P 2 P ðþ ¼ m2ð2lþm þm þm 2 2 Þ l½ðlþm Þðm þm 2 Þþm m 2 Š : We refer to Eq. (5) together with Eqs. (6) (7) as the mean value approach with exact fractions of blocked obs. On the other h, using Lemma, we can approximate the fractions of blocked obs by ~P ¼ l l ; P ~ m m 2 ¼ l : 2 m 2 We shall refer to Eq. (5) with the above approximations in place of P P 2 as the fixed point approach. We note that the fractions P P 2 have not been calculated in [4]. We have indicated there that the comparison of the exact model with the fixed point approximation is the topic of the ensuing research. We have also performed Monte Carlo simulations. First we plot the expected total soourn time of a ob in the system obtained by four approaches: the exact model, the mean value approach with exact fractions of blocked obs, the fixed point approach Monte Carlo simulations. Similarly to the scenario considered in [4], we vary m keeping the sum m þm 2 constant. One can see in Fig. 2 that the mean value approach with the exact fractions of blocked obs gives more precise results than Exact Model Mean Value Approach Fixed Point Approach Simulation Fig. 2. Expected soourn time as a function of m, given m þm 2 ¼, l ¼ m ¼ 2.

6 K. Avrachenkov, U. Yechiali / Computers & Operations Research 37 (2) P exact model P fixed point Fig. 3. Fraction of blocked obs at the first primary queue as function of m, given m þm 2 ¼, l ¼ m ¼ 2..9 P exact model P fixed point Fig. 6. Fraction of blocked obs at the first primary queue as a function of m, given m þm 2 ¼, l ¼ m ¼ P 2 exact model P 2 fixed point P 2 exact model P 2 fixed point Fig. 4. Fraction of blocked obs at the second primary queue as function of m, given m þm 2 ¼, l ¼ m ¼ Fig. 7. Fraction of blocked obs at the second primary queue as a function of m, given m þm 2 ¼, l ¼ m ¼ 2..7 Exact Model Mean Value Approach.6 Fixed Point Approach Simulation Fig. 5. Expected soourn time as a function of m, given m þm 2 ¼, l ¼ m ¼ 2. the fixed point approach. In Figs. 3 4 one can see that there is a gap between the exact values of the fractions of blocked obs their approximations obtained via the fixed point approach. In fact, the probabilities obtained by FPA approximate well the timeaverage probabilities of full queues but not the event-average fractions of blocked obs. Nevertheless, the behaviour of the T µ 5 5 Fig. 8. Expected soourn time as a function of m m 2, given m þm 2 þm 3 ¼ 5, l ¼ m ¼ 2. fractions of blocked obs is captured qualitatively well by the fixed point approach. In particular, we can see that the value of the fraction of the obs blocked in the first primary queue is not monotone with respect to the capacity of the first primary queue. As confirmed by Figs. 5 7, the fixed point approach approximates better the system performance as both capacities µ 2 5

7 8 K. Avrachenkov, U. Yechiali / Computers & Operations Research 37 (2) 74 8 µ available. We have proposed approximation procedures involving simple analytic expressions, based on mean value analysis on fixed point approach. The mean soourn time of a ob in the system the mean number of visits to the orbit queue are estimated by the MVA which needs as an input the fractions of blocked obs in the primary queues. The fractions of blocked obs are estimated by FPA. Using a benchmark example of the system with two primary queues, we conclude that the approximation works well in the light traffic regime. We have formulated a number of optimization problems such as capacity allocation problem. We note that our approach becomes exact if the blocking probabilities are fixed. 4 of the primary queues increase or equivalently the traffic load decreases. We observe from Figs. 2 5 that if one uses exact fractions of blocked obs, the mean-value analysis produces quite accurate results. From Figs. 2 5 it appears that the expected total soourn time of a ob in the system is minimized when m ¼ m 2.Wehave also performed Monte Carlo simulations for the model with three M/M// tem queues (r ¼ 3). We have varied m m 2, keeping m þm 2 þm 3 constant (see Figs. 8 9). In the case of three tem queues it appears that the minimum of the expected total soourn time of a ob in the system is achieved at the point m ¼ m 2 ¼ m 3. This is our conecture that we plan to study in the future.. Conclusion Fig. 9. Expected soourn time as a function of m m 2 (the same value levels), given m þm 2 þm 3 ¼ 5, l ¼ m ¼ 2. We have analysed networks of tem blocking queues having a common retrial queue, for which explicit analytic results are not µ Acknowledgements We would like to thank Alain Jean-Marie for his very helpful advices about Monte Carlo simulations. We also would like to thank anonymous reviewers the editor Antonis Economou for their suggestions. We also acknowledge the funding from Euro-NF Network of Excellence. References [] Artaleo JR. Accessible bibliography on retrial queues. Mathematical Computer Modelling 999;3: [2] Artaleo JR, Economou A. On the non-existence of product-form solutions for queueing networks with retrials. Electronic Modeling 25;27:3 9. [3] Artaleo JR, Gómez-Corral A. Retrial queueing systems: a computational approach. Berlin: Springer; 28. [4] Avrachenkov K, Yechiali U. Retrial networks with finite buffers their application to Internet data traffic. Probability in the Engineering Informational Sciences 28;22: [5] Bron J, Yechiali U. A tem Jackson network with feedback to the first node. Queueing Systems 99;9: [6] Collange D, Costeux J-L. Passive estimation of quality of experience. Journal of Universal Computer Science 28;4: [7] Falin GI. A survey of retrial queues. Queueing Systems 99;7: [8] Falin GI, Templeton JGC. Retrial queues. Boca Raton: CRC Press; 997. [9] Yechiali U. Sequencing an N-stage process with feedback. Probability in the Engineering Informational Sciences 988;2:263 5.