MULTIRATE RESOURCE SHARING FOR UNICAST AND MULTICAST CONNECTIONS

Transcription

1 MULTIRATE RESOURCE SHARING FOR UNICAST AND MULTICAST CONNECTIONS Khaled Boussetta and Andre-Luc Belyot Laboratoire PRiSM University o/versailles Saint-Quentin 45, Avenue des Etats-Unis Versailles, France Abstract Keywords: Recent advances in network equipment and the important progress in video, audio and data compression techniques has given rise to new multimedia group applications, such as Video-conferencing or Webcasting. These applications need guarantees on the QoS (Quality of Service) and require group communication services. ATM seems to be well adapted to support these applications, since it provides several service classes and QoS guarantees. However, these guarantees are possible only if the network is able to allow efficient resource sharing. Therefore, resource sharing for multirate traffic is a fundamental issue, specially for multicast connections with a huge number of recipients. In this paper, we generalize the mathematical model of Kaufman [6] to compute the exact values of call blocking probability for unicast and multicast multirate multiclass traffics sharing a single link. We show that the model has a product form solution. Moreover, we give a recursive algorithm to obtain the normalization constant and the call blocking probability. Multicast, resource sharing, product solution, recursive solution, ATM. 1. INTRODUCTION Over the last few years, several multimedia group applications have emerged. In particular, the extension of the Internet to support the multicast within the MBONE [2] makes the use of multimedia group applications over wide area networks a commonplace reality. Some of these applications, such as CSCW (Computer Supported Cooperative Work) or video-conferencing, are very popular within the Internet community. These applications facilitate group communication and allow members of a group to work together or to share resources. We believe that there will be an increasing need for developing this kind of applications in the future and so, there will be an absolute necessity to provide the D. H. K. Tsang et al. (eds.), Broadband Communications Springer Science+Business Media New York 2000

2 562 multicast capability in the networks. The support of multicast leads to many problems [4] such as call admission control, multicast routing [3], group management, multicast reliability [1], flow control [8], etc. However all these questions are a consequence of one fundamental problem: the resource sharing problem. Until now, most of the answers given to these problems aim at supporting unicast connections with heterogeneous requirements. Thus, multirate multiclass loss models for unicast traffic, have been widely studied. It was shown that they have a product form solution, and exact blocking probability can be computed. Moreover, recursive solutions have been developed to bypass numerical problems that may arise in large systems. Efficient Sharing of resources for multicast connections is required to allow the support of a huge number of recipients. However, the performance of a link supporting multicast connection has not been extensively investigated. Therefore, most of these research works concern the switch call blocking probability [7]. [5] gives an interesting solution to compute multicast call blocking in a single link, that is, the authors use generalized Engset system to deduce the call blocking probability. Nevertheless, only multicast traffics have been considered, while the effect of unicast connections sharing the same link was not discussed. Moreover, difficulties in computing will arise for large systems since no recursive solution is given. In this paper, we use a mathematical model to compute the exact solution of the call blocking probability for both unicast and multicast multirate traffics sharing a single link. Our work differs from [5] in that we generalize the model of Kaufman [6] to compute the exact steady-state blocking probability for multicast and unicast multi rate multiclass connections. To obtain the normalization constant, we use a new recursion algorithm which gives the resource distribution for multicast connections. Finally, the call blocking probabilities for multicast and unicast traffics are obtained with our recursive algorithm. The rest of the paper is organized as follows. In the next section, we present the problem of resource sharing for unicast and multicast traffic, then we present our analytical model and its product solution form. We also describe our recursive solution to compute the call blocking probability. In section 3, an example is given to illustrate the use of our recursive algorithm. A summary of this paper and possible extensions are discussed in section 4.

3 CALL BLOCKING FOR MULTIRATE TRAFFIC 2.1 MODEL DESCRIPTION We study the call blocking probability in a link with C resource capacity that allows the support of unicast and multicast multirate multiclass streams. All arrival call connections and holding times are independent. We suppose that the other links in the network have infinite capacity, so that no blocking that may affect our system occurs. Let U be the number of unicast customer types. Class i unicast customers arrive according to a Poisson process of rate Au,i and require simultaneously bu,i resource units. We assume that the holding time of class i call is exponentially distributed with mean _1_. Thus, the offered load is Pu i = ~ A Unicast P-u, ' J..tu,i customer whose requirement cannot be satisfied is blocked. Suppose that there is M multicast customer types, each class corresponds to a multicast session. Thus, an infinite user population can join each multicast session. That is, it is reasonable to consider that multicast customers of class i arrive according to a Poisson process with rate Am,i and has exponential residency time with parameter j.lm,i. Thus, the offered load is Pm,i = >'::,;:.. All the members of a given multicast session will require the same bandwidth capacity. That is, multicast customers of class i, will require bm,i resource units. Definition 1 i is an active class if there is at least one multicast customer of class i being served in the link. As calls of multicast class i will not be blocked if i is already active, then bm,i units of the system are allocated to a new arriving multicast customer of class i, only if i is not already an active class and there is at least bm,i free resources in the system. 2.2 PRODUCT FORM SOLUTION Under the preceeding assumptions, it is easy to see that the state of our system is described by the state vector n = (u, m) = (Ul,, Uu, ml,, mm). Here u is the vector number of calls in progress for unicast class and m is the state vector for multicast ones. Let n be the set of allowed states. n depends on the sharing policy P. In this paper, we address Complete Sharing policy (CS). However, join distribution will also have a product form solution for Upper Limit (UL) and Guaranteed Minimum (GM) policies.

4 564 Let l{mi>o} = 1 if i is an active class. Then, the set of allowed states 0 for CS policy can be written as: { U M 0= (u, m) / ~ bu,i' Ui + ~ bm,i.1{mi>o} (1) Theorem 1 The state probability corresponding to a resource sharing policy CS is given by : 1 U ti, M m, P () Il Pu,i Il Pm,i u,m = G(O). -.,' -., i=l Ut i=l m t (2) Where G (0) is the normalization constant for O. U u, M m,) G (0) = G (C, U, M) = 2: ( Il P~'i. Il Pm.',i U t. m t nen i=l t=l The proof is trivial since the state probability verifies the global equilibrium balance equation. 2.3 CALL BLOCKING PROBABILITY Let Bt denotes the sets of blocking states for unicast or multicast class i. Then, the call blocking probability for class i can be expressed as : (3) Pbi = 2: P(n) = 1-2: P(n) = 1-2: P(n) (4) Thus, (5) Furthermore, when i is a unicast class, then Kaufman [6] shows that the previous quantity can be expressed as : R. = 1 _ G (C - bu,;, U, M) bt,u G (C, U, M) (6)

5 565 Let M - i, denotes M multicast classes without the class i, which also means that the class i will never be active. Then, a similar expression for the multicast call blocking probability Pbi,m can be found. See the Appendix for the proof. epm,i G (C - bm,i, U, M - i) Pbi,m = 1- G (C, U, M) (7) 2.4 UNICAST CONNECTIONS In the previous section, we gave the exact solution to compute the call blocking probability for both multicast and unicast classes. However, the state system has as many dimensions as the number of traffic classes. Therefore, in a large system capacity C and a huge number of classes M + U, numerical difficulties may arise when computing both the normalization constant and the call blocking probability. Fortunately, Kaufman [6] found a solution to map the multi-dimensional state space into a one dimensional state space. It has been shown that the distribution of the number of resource units allocated for unicast connections has a recursive expression. Moreover, this formula is only linearly dependent on U. These results can also be applied in our model to compute the resource distribution for unicast connections. In fact, since the state probability has a product form, it can be easily verified that the multicast traffic has no consequences on the validity of the recursive solution. However, since the normalization constant in our model depends on both unicast and multicast connections, we have to consider first an unormalized state probability for resource distribution for the unicast traffic. Thus, let qu (.) be this quantity. Then: (8) Here j denotes the number of resource units allocated to the unicast connections and u.bu = L~l bu,iui. Kaufman [6] shows that qu (j) can be computed recursively as follow:,j < 0,j = 0,j> 0 (9)

6 MULTICAST CONNECTIONS The recursive solution given in the previous section cannot be applied for the multicast connections. However, in the last case, we find another solution to reduce the state space. Thus, let us define q M (.) as the unormalized state probability for resource distribution in the multicast traffic. Remembering that a class i request will never be blocked if i is an active class, an infinite population customer of the same class i can join an active class i. Thus, defining 1m.bm = 2:f!1 bm,i.1{mi>0} we can write: Thus, we find qm (j) = L II l{m.>o} {m:l m.bm=j} i=1 M ( +00 pm,.) """'~ ~ m' m,=1,. (10) qm (j) = L II l{m,>o} (e Pm., - 1) M {m:lm.bm=j},=1 (11) Now let qk (.) be the unormalized resource distribution probability for multicast classes whose index vary from of I to k. Let b~l denotes a k dimension capacity vector for multicast calls. Thus, 1~.b~n = 2::7=1 bm.i.1 {m, >O} k II l{m.>o} (e Pm. - 1) (12) Since this quantity can be divided into two parts, the first one represents the case where no customers of class k are being served, and the second one for the case where class k is active, then: qk (j) = 2::{m:l~.b~=jlmk=0} I17=1 l{m.>o} (e Pm. - 1) + 2::{m:l~.b~=jlmk>0} I17=1 l{m.>o} (e Pm., - 1) (13) Consider the left-side of the sum of equation 13, then this quantity can be written as follows: { 'lk-l bk - 1 m. m. m -} _ '} i=1 k-l II l{mi>o} (e Pm., - 1) = qk-j (j) (14)

7 567 The right-hand side of the sum of equation 13 can be written as follows: k-l (epm,k - 1) II l{m.>o} (e Pm,. - 1) { 'lk-l bk - 1 m. m. m -J _ '-b m,k } i=1 (15) We find the recursive formula for the multicast traffic:,j < O,Vk,j = O,Vk,j > 0, k = 0,j > 0, k > 0 (16) Finally, using 9 and 16 we can compute the normalization constant G (f2) : G (Il) = t. qm (i) (~qu (j)) (17) 2.6 MULTICAST CALL BLOCKING PROBABILITY A multicast call of class r is blocked only if r is not an acti ve class and there are less than bm,r free resource units in the link. Thus, let qm -r (.) be the unormalized resource distribution probability for multicast connections such that the class r is not active: M qm-r (j) = II l{m.>o} (epm" - 1) (18) Gi ven the recursi ve formula gi ven in 16 we obtain the following: Thus we have a recursive formula that will give the exact value of qm -r (j) { 0 qm-r (j) = 1 qm (j) - (epm,r - 1) qm-r (j - bm,r),j < 0,j = 0,j > 0 (20) Finally, the call blocking probability for unicast and multicast connections can be computed using the following equations:

8 568 fl,i,m = G ~n) t, (qu (j) ~ qm-i (C - j - k)) (22) 3. EXAMPLE We consider a link with a bandwidth capacity equal to 10 Mbps. Let the basic bandwidth unit be 64 Kbps. Thus, C = 160. We choose two unicast classes: U = 2 and four multicast classes: M = 4. For each class we take the following capacity requirements: bu,l = 32, bu,2 = 16A bm,k = 32, 'tk/1 :S k :S M We suppose that for both unicast and multicast mean call duration time are 120 seconds. Thus, to analyze the link occupancy at different loads, we introduce A as the inter-arrival time of a customer (unicast and multicast) in the link such as: Au,l = A Au,2 = 2A Varying A from 0 to 900, we compute using equations 21 and 22 the call blocking probability for unicast and multicast traffics. The results are plotted in figure 1. The figure shows that the call blocking probabilities for unicast traffics increase when the traffic load grows. However, multicast call blocking probabilities for all classes are always under a certain threshold. Thus, these probabilities increase until a certain value of A, after which they decrease. This is due to the fact that after a certain value of A, these classes will always be active. At this point, the traffic load is so high that there is at any time at least one multicast customer being served; thus, for these multicast sessions a new customer will never be blocked.

9 569 Unicast > Q..Q " 0. '" C... u '"..... u 0.2 " Lambda Figure J Call Blocking Probability 4. CONCLUSION In this paper, we addressed the multirate resource sharing problem for unicast and multicast traffics. We show that in Complete Sharing policy the model have a product form solution. Thus, we deduce the exact call blocking probability values for both unicast and multicast connections. Moreover, to avoid the numerical problems that may arise when computing these quantities, we extend the recursive solution of Kaufman [6] to the multicast case. Our future prospects will concern the adaptation of these results to other sharing policies, such as Upper Limit (UL) and Guaranteed Minimum (GM) policies. We also investigate the problem of multicast resource sharing in a wireless environment. Appendix Let us write G (n - Bn. This quantity can be divided into two parts, the first one represents the case where there is no customer of class i being served, thus, no blocking occurs for this class if the sum of the allocated resource units in the system is under C - bm,i' The second part of the equation represents

10 570 the case where the class i is active, that is no blocking for this class will occur. Therefore, We obtain that: G ( f"l _ Bt) =,",C-bm.i (.) (,",C-b m. - j (k)) H L..Jj=o qm-i J L..Jk=O qu + (e Pm, - 1) L:T::b m. qm-i (j - bm,i) (L:~==-6 qu (k)) (A. I) Thus, taking J = j - bm,i in the second sum of A.I we find that: G (0 - Bt) = e Pm, G (C - bm,i, U, M - i) (A.2) We can now deduce that the multicast call blocking probability for i is : epm G (C - bm,i, U, M - i) Pbi,m = 1- G (C, U, M) (A.3) References [1] C. Diot. Reliability in multicast services and protocols; a survey. In International Conference on Local and Metropolitan Communication Systems, December [2] Hans Eriksson. MBone: The Multicast Backbone. Communications of the ACM, 37:54-60, [3] D, Reeves H. Salama and Y. Viniotis. Evaluation of multicast routing algorithms for real-time communication on high-speed networks, IEEE Journal on Selected Areas in Communication, February [4] G. C. Polyzos J. C. Pasquale and G. Xylomenos. The multimedia multicasting problem. ACM Multimedia Systems Journal, 6(1 ):43-59, [5] S. Aalto J. Karvo, J. Virtamo and O. Martikainen. Blocking of dynamic multicast connections in a single link. In International Broadband Communications Conference. Future of Telecommunications, pages Paul J. Kuhn and Roya Ulrich Edition, April [6] D. Kaufman. Blocking in a shared resource environment. IEEE Transactions on communications, 29(10): , October [7] J. S. Turner. An optimal nonblocking multicast virtual circuit switch. Technical Report wucs-93-30, March [8] H. A. Wang and M. Schwartz. Performance analysis of multicast flow control algorithms over combined wireless/wired networks. In IEEE INFO COM'97, April 1997.