Efficient Importance Sampling for Monte Carlo Simulation of Multicast Networks

Transcription

1 Efficient Imortance Samling for Monte Carlo Simulation of Multicast Networks P. Lassila, J. Karvo and J. Virtamo Laboratory of Telecommunications Technology Helsinki University of Technology P.O.Box 3000, FIN-025 HUT, Finland {Pasi.Lassila, Jouni.Karvo, Abstract We consider the roblem of estimating blocking robabilities in a multicast loss system via simulation, alying the static Monte Carlo method with imortance samling. An aroach is introduced where the original estimation roblem is first decomosed into indeendent simler sub-roblems, each roughly corresonding to estimating the blocking robability contribution from a single link. Then we aly imortance samling to solve each sub-roblem. The imortance samling distribution is the original distribution conditioned on that the state is in the blocking state region of a single link. Samles can be generated from this distribution using the so called inverse convolution method. Finally, a dynamic control algorithm is used for otimally allocating the samles between different sub-roblems. The numerical results demonstrate that the variance reduction obtained with the method is remarkable, between 400 and in the considered examles. Keywords Multicast, loss systems, simulation, Monte Carlo methods, variance reduction, imortance samling. I. Introduction We consider the calculation of blocking robabilities in treestructured multicast networks with dynamic membershi. In these networks, users at the leaf nodes can oin or leave any of the several multicast channels offered by one source (i.e. the root of the tree) in the network. The users oining the network form dynamic multicast connections that share the network resources. Blocking occurs when there are not enough resources available in the network to satisfy the resource requirements of a request. Since blocked calls are lost, this system is also called a multicast loss system. This scheme is alicable to circuit switched systems as well as acket switched systems with strict quality guarantees for multicast flows. Karvo et al. [1] studied this system under the assumtion of having an infinite user oulation generating call requests at the leaf nodes and a single finite caacity link in the network. An exact algorithm to comute the blocking robabilities was derived. This work was extended by Boussetta and Belyot [2] by adding unicast traffic to the system. Reduced load aroximations of the blocking robabilities in a network were derived in [3]. An exact algorithm for the network case has been given in Nyberg et al. [4]. A roblem with the exact solution, however, is that it cannot be comuted for networks with a large number of channels, I, due to the exonential growth of the size of the state sace; the comlexity of the algorithm is of order O(2 2I ) (however, the comlexity grows only linearly with resect to the number of links in the network). Therefore, to be able to analyse systems with a larger number of channels, new methods need to be develoed for estimating the blocking robabilities quickly and reliably. One ossible aroach is to use simulations. As the form of the stationary distribution is known, the static Monte Carlo (MC) method can be used. In order to make the simulation more efficient, it is ossible to use imortance samling (IS), where one uses an alternative samling distribution, which makes the interesting samles more likely than under the original distribution. The twist in the distribution is then corrected for by weighting the samles with the so called likelihood ratio. The use of IS in MC estimation of blocking robabilities has been reviously studied in [5], [6], [7], and [8]. However, the articular loss system that has been studied in these works is the so called multiservice loss system. The multicast network studied here is in many ways different from the multiservice loss system, but it ossesses sufficiently common features with the multiservice loss system to allow us to aly in this case the so called inverse convolution method develoed in Lassila and Virtamo [8]. In this aer we first show how the roblem can be decomosed into indeendent sub-roblems. The decomosition corresonds to breaking the blocking robability down into comonents each of which essentially gives the blocking robability contribution from a single link. Then we resent an efficient IS distribution to be used to estimate the blocking robability contribution from each link. The distribution is a conditional distribution and allows one to generate samles directly into the set of blocking states of a given link, assuming that link solely to have a finite caacity. To generate these samles we use the inverse convolution method. We will show via numerical examles the efficiency of the above method when comared with a direct Monte Carlo aroach. The aer is organized as follows. Section 2 resents briefly the multicast loss system and the decomosition aroach. In section 3 we discuss the efficient estimation of blocking robabilities when alying the decomosition and imortance samling. Section 4 contains the main result of the aer, describing the inverse convolution method for multicast loss system. In section 5 we describe the dynamic method for otimally allocating the number of samles to be used for each sub-roblem and give some numerical examles demonstrating the effectiveness of the method. Section 6 contains our conclusions //$ IEEE 432 IEEE INFOCOM 20

2 II. The multicast loss system We define the multicast loss system in the same way as Nyberg et al. [9]. Consider a network consisting of J links, indexed with =1,...,J, link having a caacity of C resource units. The set of all links is denoted by J. The network is organized as a tree, where the root link is denoted by J. The set U denotes the set of user oulations, located at the leaves of the tree. The leaf links and the user oulations connected to them are indexed with the same index u U= {1,...,U}. The set of links on the route from user u to the root node is denoted by R u. The user oulations which use link, i.e. for which link R u, are denoted by U. The size of the set U is denoted by U. The multicast network suorts I channels, indexed with i I= {1,...,I}. The channels originating from the root node reresent different multicast transmissions, from which the users may choose. Each channel has a caacity requirement d i d = {d i ; i I}. We assume the d i and C to be integer multiles of a basic resource unit. In this work, we use the same caacity requirement for all links, but it is trivial to generalize this model to link deendent caacity requirements, by using d,i. The state of a link is defined by the states of the channels i. Each channel may be either on (1) or off (0), i.e. the state of channel i on link is Y,i {0, 1}. The state of a link is denoted by the vector Y = {Y,i ; i I} S, where S is the link state sace S = {0, 1} I. A multicast connection is defined by the air (u, i) of the user u (leaf link) and channel i. Synonymously, we refer to the multicast connection (u, i) as a traffic class. The network state X is defined by the states of the leaf links Y u S, X =(Y u ; u U)=(Y u,i ; u U,i I) Ω, (1) where Ω={0, 1} U I denotes the network state sace. The state of link is determined by the network state as follows: Y u if = u U, Y = Y u otherwise, (2) u U where denotes an OR-oeration between channel states in different links, i.e. channel i is on on link if and only if there is a link u U on which the channel is on. We denote the link caacity occuancy (or link occuancy for brevity) of link by S, I S = d Y = d i Y,i. Now, in a finite caacity network, the state sace is truncated by the caacity constraints of the links, { } Ω = x Ω S C, J. i=1 A. Probability distributions and blocking Let us assume that in a system with infinite link caacities, the user oulations of the leaf links are indeendent, and that the leaf link distributions π u (y u ) = P{Y u = y u },u U are known, and reresent stationary distributions of reversible Markov rocesses satisfying the detailed balance equations. Several tyes of user oulation models of this kind have been discussed in [9]. In our work, we use a user oulation model for which traffic classes (u, i) are indeendent so that π u (y) = i I yi u,i (1 u,i) 1 yi, (3) where u,i =P{Y u,i =1} is the robability that channel i is on on leaf link u. The robability,i of channel i to be in the on state on any link may be calculated from the corresonding known robabilities of the leaf links U,,i =1 u U (1 u,i ). An examle of this kind of a rocess might be an infinite oulation of users generating calls (connections to the root of the tree) as a Poisson rocess with intensity λ u. Arriving calls select channel i with a robability α i. The channel onrobability can be calculated by u,i = 1 ex( λ u α i τ i ), where τ i denotes the mean holding time of channel i calls. Note that the steady state distribution is insensitive to the call holding time distribution. The steady state robabilities π(x) of the network states in a system with infinite link caacities can then be calculated from π(x) =P{X = x} = u U π u (y u ), since the user oulations are indeendent. Due to the assumed detailed balance, the robabilities π(x), x Ω, of states in a system with finite link caacities are now obtained simly by truncation, π(x) =P{X = x X Ω} π(x) = P ( Ω), x Ω, 0, otherwise, where P ( Ω) = P{X Ω} = x Ω π(x). In a finite caacity network, blocking occurs when a user tries to establish a connection for channel i, and there is at least one link R u where the channel is not already on and there is not enough sare caacity for setting the channel on. Without loss of generality, we assume that the channels are ordered so that the channel for which we are calculating the blocking robability has the greatest index I. With this convention, the channel index is unnecessary and will be omitted for most of the notation. Let S denote link occuancy due to the channels {1,...,I 1}, i.e. I 1 S = d i Y,i. i= //$ IEEE 433 IEEE INFOCOM 20

3 Then, the set B u of states where connections of traffic class (u, I) are blocked is defined as { B u = x Ω } R u : S >C d I,. We use the exression link blocks if for link, S > C d I. Thus the set B u consists of the states where at least one link blocks. Then the time blocking robability for traffic class (u, I) is The sets Du and Eu are illustrated in Figure 1. The real state sace is imossible to draw, since it has U I dimensions, and has only two oints er each dimension. Thus, the figure only illustrates the rincile of decomosition showing an examle where the sets consist of elements having two comonents (x 1,x 2 ). In the figure, B u is reresented by the grey area and it consists of the union of three disoint subsets Eu 2, E3 u (light grey areas) and Eu 1 (dark grey area). Also note that each Eu is a subset of the corresonding Du. B u =P{X B u X Ω} = P{X B u} P{X Ω} = P (B u) P ( Ω), (4) where P (B u )=P{X B u }. x 1 E 3 u D 1 u D 2 u D 3 u B. Decomosition In order to divide the task of estimating P (B u ) to simler sub-roblems, we artition B u into sets Eu. Eu is defined as the set of oints in B u where link blocks but none of the links closer to user u block, { Eu = B u x Ω S >C d I } S C d I, R u, where R u denotes the set of links on the ath from u to, including link u but not link. The Eu obviously form a artitioning of B u, i.e. B u = Eu, R u and E u E u =, when. From this it follows that P{X B u } = R u P{X E u}, (5) and we have decomosed the roblem into simler subroblems of determining the robabilities P{X Eu}. The robability P{X Eu} can be thought of as the blocking robability contribution due to link. It should be noted, however, that blocking in the states where several links block can be arbitrarily attributed to any of the blocking links. We have adoted the convention which attributes it to the blocking link closest to the user. For later use, we introduce the suerset Du Eu, which will have an imortant role in imortance samling, { } Du = x Ω C d I <S C Y,I =0. This set corresonds to blocking states in a system where link has a finite caacity C but all other links have infinite caacity. However, in real systems, all links have finite caacity, and several links could block simultaneously. Thus, sets Du are not disoint unlike their subsets Eu. B u E 1 u E 2 u Fig. 1. Decomosition of B u into sets D u and E u. III. Efficient imortance samling In what follows we discuss the efficient estimation of the blocking robabilities. As the form of the stationary distribution π(x) is known, a natural choice for the simulation method is the static Monte Carlo method. The main roblem in the simulation is to quickly get a good estimate for P{X B u }, i.e., the numerator in Eq. (4), esecially in the case when the blocking robability B u is very small. For comleteness, recall that B u also deends on P{X Ω} given by the denominator of (4). This robability is usually close to 1 and is easy to estimate using the standard MC method. Therefore, in the rest of this aer we concentrate on efficient methods for estimating P{X B u }. As already noted, Eq. (5) allows us to decomose the estimation of P{X B u } into indeendent sub-roblems of estimating the P{X Eu }. Now, the idea of our imortance samling method is simly based on exressing P{X Eu } as a conditional robability, x 2 P{X E u} =P{X E u X D u}p{x D u}. (6) This relation is useful from the simulation oint of view since we can comute P{X Du} exactly (see [1]) and we can efficiently generate oints from the original distribution under the condition X Du, as exlained later. Then we only need to estimate via MC simulation the conditional robability =P{X Eu X D u } instead of =P{X Eu} (see Figure 2). The estimation of is much more efficient than the estimation of since tyically is much greater than. The efficiency gain obtained with the above can //$ IEEE 434 IEEE INFOCOM 20

4 be shown as follows. When estimating via standard MC simulation each samle is an indeendent Bernoulli variable and the relative error (or relative deviation) of the estimate, given by the ratio of the standard deviation and the mean of the estimate, after N samles have been drawn is (1 )/(N). Similarly, when estimating the relative deviation is given by (1 )/( N) Tyically, the blocking robabilities are of the order 1% and to illustrate the efficiency gain we can consider e.g. an examle where = Then we need almost samles to get a relative error of 5%. On the other hand, when estimating we need less samles to reach the same accuracy level. How much less deends on how big a art of E u is inside D u. Tyically is in the range 0.5,...,1. Assuming, e.g. that =0.9, we only need about 45 samles to reach the same 5% relative error level, giving us a decrease by a factor of almost 2000 in the required samle size. Our numerical results also indicate that variance reductions of this order can indeed be obtained. x 1 D 1 u Fig. 2. E 1 u Ω Estimation of P{X E u X D u}. From Eq. (6) we have the following estimator, η u, for η u = P{X E u}, η u = v u N N x 2 1 X n Eu, (7) n=1 where vu =P{X D u } and X n denotes samles drawn from the conditional distribution (x) =P{X = x X D u}. Observe that Eq. (7) corresonds to the same estimator for ˆη u if we take the conditional distribution (x) as our imortance samling distribution in the imortance samling estimator η u = 1 N 1 N X n Eu w(x n), n=1 where w(x) =π(x)/ (x) =v is the so called likelihood ratio, the value of which in our case is constant. Finally, the estimator for P (B u ) is simly P (B u )= R u η u. Given the total number of samles N to be used for the estimator, the number of samles N allocated to each sub-roblem is a free arameter. In section V-A we show how to choose each N to minimize the variance of P (B u ). IV. Inverse convolution method In this section, we resent the inverse convolution method (IC) for samle generation. We are now only considering the estimation of one ηu for fixed R u and traffic class (u, I). The following method is based on the observation that it is relatively easy to generate oints from the conditional distribution (x) =P{X = x X D u } by reversing the stes used to calculate the occuancy distribution of the considered link. Note that the condition X Du is a condition exressed in terms of the occuancy, S, of the considered link. The idea in the inverse convolution method is to first generate a samle of Y such that the occuancy of the link is in the blocking region. Then, given the state Y, the state of the network, i.e. states of the leaf links, is generated. The maing : X Y is surective, having several ossible network states X generating the link state Y, and we draw one of them according to their robabilities. The main stes of the simulation can be summarized as follows: 1. Generate the states for leaf links u by (a) Generate a samle state Y under the condition C d I < S C Y,I =0for link. (b) Generate the leaf link states Y u, u U, with the condition that link state Y = u U Y u is given. (c) Generate the states Y u, u U U for the rest of the leaf links as in the normal Monte Carlo simulation. 2. The samle state of the network X n Du consists of the set of all samle states of leaf links generated with ste To collect the statistics for estimator (7), check if X n Eu. The above stes are reeated for generating N samles. The method of generating a samle for link (ste 1a) is exlained in more detail in section IV-A. The method for generating the leaf link states from the link state (ste 1b) is exlained in section IV-B. See figure 3. A. Generating a samle for D u As already noted, we have artitioned the set of blocking states into disoint sets Eu. However, it is not easy to generate samles directly to these sets. Instead, we generate samles to sets Du which corresond to the states in which at least link blocks. After that it is ossible to check if the samle belongs to the set Eu to collect the sum in Eq. (7). First, the link occuancy S is easily calculated recursively as follows. Let S,i denote link occuancy due to the first i channels, S,i = i i d i Y,i. Then S = S,I and S = S,I 1. The Y,i are mutually indeendent, and we can exress S,i = S,i 1 + d i Y,i, where //$ IEEE 435 IEEE INFOCOM 20

5 O Fig. 3. Examle of generating a samle in the set Du. First, a samle state for the link, denoted by the thick dashed line, is generated by inverse convolution. Given that channel i is on in this state, the states of that channel are generated for the links marked by the dashed ellise by another inverse convolution ste (if channel i is off on link then it is off on all those links). This is reeated for all i. States for the links denoted by ticks are generated by a simle draw. S,i 1 and Y,i are indeendent. For samle generation, we are only interested in the occuancy generated by the first I 1 channels, S, since a call for a channel cannot be blocked if it is already in the on state. This is also reflected in the definition of the set Du. Let P{S,i = x} = q,i(x) denote the robability distribution of S,i. Now, the robability mass v of the set Du, can be calculated as v =P{X D u } =(1,I) C i=c d I+1 u q,i 1 (i). The link occuancy distribution q,i 1 may be calculated recursively by convolution: q,i (x) =q,i 1 (x d i ),i + q,i 1 (x)(1,i ), (8) where the recursion starts with q,0 (x) =1 x=0. For interretation of the convolution ste, note that the event {S,i = x} is the union of the events {Y,i = y,i,s,i 1 = x d i y,i }, y,i {0, 1}, where y,i = 0 means that the channel i is in the off state, and y,i =1means that the channel i is in the on state on link. The corresonding robabilities are q,i 1(x)(1,i ) and q,i 1(x d i ),i, resectively. Conversely, we can infer what is the conditional robability of the event {Y,i =1,S,i 1 = x d i } given that S,i = x, P{Y,i =1,S,i 1 = x d i S,i = x} = q,i 1 (x d i ),i q,i 1 (x d i ),i + q,i 1 (x)(1,i ) = q,i 1 (x d i ),i. q,i (x) The robability of the event {Y,i =0,S,i 1 = x S,i = x} is then 1 P{Y,i =1,S,i 1 = x d i S,i = x}. Having all the necessary tools, we are now able to generate samles to the set D u. This set corresonds to states in which C d I <S C and Y,I =0. Generation of a state starts (9) by drawing a value for S = S,I 1 using the distribution q,i 1 ( ) with the condition that C d I <S C. This conditional distribution can be recomuted and stored. Then, given the value of S,I 1, the state Y,i of each channel (i = I 1,...,1) is drawn in turn using robabilities (9). Concurrently with the state Y,i, the value of S,i 1 becomes determined. This is then used as the conditioning value in the next ste to draw the value of Y,i 1 (and of S,i 2 ), etc. Drawing a samle for Y,i requires ust generation of a uniform random number from the interval (0,1) and setting the channel to on state if the number is smaller or equal than the robability. Note that for reasonable sizes of links, it is advantageous to store the robabilities for fast generation of samles. The next subsection resents a method for drawing leaf link states Y u, given the state Y of link. B. Generating leaf link states from a link state Having drawn a value for state Y of link, it is ossible to draw values of the state vectors Y u, u Uof the leaf links. For u U, states Y u are generated under the condition Y = u U Y u using a similar inverse convolution rocedure as above. This condition can be broken down into searate conditions for each channel, i.e. for each i we have a searate roblem of generating the values Y u,i, u U, under the condition Y,i = u U Y u,i with a given Y,i. Note that only channels i which are in the on state on link, Y,i =1, need to be considered. If Y,i =0, then necessarily Y u,i =0for all u U. The above conditions affect leaf links u U. For other links u U U, the states Y u are indeendently generated from the distribution (3). First, let us consider a convolutional aroach for generating a link state for channel i and link if we already know the states for each link u U. In this section, we use an index u {1,...,U } = U for the subset of leaf links. Let S u,i = x denote the event that the channel is on state x on link when u =1,...,u leaf links have been counted for, i.e. S u,i = u u y u,i. Let P{S u,i =1} = q u,i be the robability that the channel is on in at least one of the links {1,...,u } U. These robabilities can be calculated recursively as follows: q u,i =1 (1 u,i)(1 q u 1,i) =(1 q u 1,i) u,i + q u 1,i. The recursion starts with q 0,i =0. If S u 1,i =1, then necessarily S u,i =1in any case, see figure 4. Conversely, to generate the state for each leaf link, given the value of Y,i,we first check if the channel would be off after u 1 leaf links counted for, since if it is, then it has to be on on leaf link u, and off on the rest of the links, which haens with robability P{S u 1,i =0 S u,i =1} = (1 q u 1,i) u,i = (1 q u 1,i) u,i. (1 q u 1,i) u,i + q u 1,i q u,i //$ IEEE 436 IEEE INFOCOM 20

6 Note that the event S u 1,i =0imlies directly that {Y u,i = 0, u <u }. The robability P{S u 1,i =1 S u,i =1} is given by 1 P{S u 1,i =0 S u,i =1}. In the case of the event S u 1,i =1, we need to check if the channel is on on the leaf link u, which haens with robability P{Y u,i =1 S u 1,i =1} = u,i. Thus we generate a uniform random number first to decide if the leaf link is the last one for which the channel is on. If it is not, we generate yet another uniform random number to decide if the channel should be set on. on off Fig. 4. Convolution aroach for generating link state from leaf link states. Each link is convoluted in turn. Note that if the channel is already on, it will be on in the next ste also, indeendently of the state of the convoluted link. This rocedure is reeated for each channel. As a result of this rocedure, we have the state vectors of each leaf link u U. The rest of the leaf link states must be generated as in the normal Monte Carlo simulation using Eq. (3). C. Comlexity The generation of samles is almost as fast as in the standard MC method, once the conditional distributions have been comuted. Generation of the link state in Du is as fast as generating a standard link state. Generating leaf link states takes 2(U 1) stes at maximum, comared with U stes of the standard MC method (generating U leaf link states). Thus, the worst case time to generate a samle state with the roosed method is twice that of the standard MC method. Furthermore, the memory requirements of the algorithm, i.e. the number of elements in the arrays, are not rohibitive. The number of array elements to be stored can be seen to be I(C + U). It should be noted that the deendence on I and U is only linear, in site of the exonential growth in I of the state sace Ω. Our method scales very favourably with the network size, essentially defined by the number of user oulations U. Aarently, the decomosition leads to relication of the simulation for each link on the route of the connection. Per each subtask, the work is about the same as in standard MC. The number of links on the route, however, grows relatively slowly, and in ractice will never be very large. Moreover, otimal allocation of samles between the subtasks, discussed in the next section, essentially eliminates the slight disadvantage. V. Numerical results A. Allocation of the samle oints We also imlemented and tested a scheme of allocating samle oints otimally for each estimation of η u, as exlained in [8]. The inverse convolution method with samle allocation (ICSA) is resented in this subsection. The η u for different links have differing variances, and contribute differently to the total variance of the estimate P (B u ). Thus, we may vary samle size for different links to minimize the total variance. The otimal allocation of samles is N = s J i=1 s i N, =1,...,J, (10) where we have denoted s 2 =V[1 X E u ]=(1 )/, where =P{X E u}. Of course, the s are not known before the simulation. Therefore a dynamic samle allocation scheme is needed. One ractical solution is to make the simulation in batches, using J M samles er batch, where M is a suitable integer, for instance M = 100. In the first batch, all the samles are distributed evenly between different links, i.e., M samles are used er link. Then initial estimates for the s are obtained. Using these estimates, the otimal samle sizes after the second batch, i.e. for N =2J M, can be calculated from (10). If the calculated N is less than the number of samles already used (M samles in the first batch) no samles of the new batch are allocated for that link. Otherwise, the available J M new samles are distributed between the links in roortion to the deficiencies (deficiency being the difference between the calculated otimal value after the new batch and the actual number of samles used so far). Real numbers are aroriately rounded to integers. After the new batch, new estimators are calculated for the s and the rocedure is reeated. B. Numerical examles Here some numerical examles are resented in order to illustrate the efficiency of the resented method in Monte Carlo simulation of the blocking robabilities. We consider the same network used in [4], [9], for which we know the exact results. The network is shown in figure 3. There is a root node, eight channels, I =8, with d i =1for all channels. The caacity of the root link is C J =7, for the others, C =6. Each leaf link has an infinite user oulation offering traffic to each channel with intensity aα i, where α i comes from the truncated geometric reference distribution with =0.2. We simulated blocking for channel I with three values for a: 1.0, 1.3 and 2.0 to comare the simulation methods in light load, moderate load and high load circumstances. To this end, we estimated the relative deviation of the estimator, given by (V[ P (B u )]) 1/2 / P (B u ), for 10 4 samles (Case 1) and 10 5 samles (Case 2). For classic Monte Carlo (MC), the total number of samles were used, while for Inverse Convolution method (MC-IC), one third of samles was used for each //$ IEEE 437 IEEE INFOCOM 20

7 TABLE I The relative deviation of the estimates P (B u) for the first examle. Case 1: samles, Case 2: samles. Case a / B u relative deviation MC MC-IC MC-ICSA 1.0 / 0.56% / 1.7% / 7.2% / 0.56% / 1.76% / 7.2% estimate η u,i. For Inverse Convolution with Samle Allocation (MC-ICSA), the total number of samles were allocated for each estimate, according to the algorithm. In these examles, we imlemented MC-IC so that it first generates a network state for estimating P ( Ω). Then it reuses this state artially for those links not generated with inverse convolution. This induces some deendence between the estimates η u in the simulation, but its effect on the variance of the estimator is not too high for these networks. For MC-ICSA this scheme of reducing rocessing time er samle is not feasible. As can be seen, the variance reductions obtained with the inverse convolution method are remarkable. For examle, for light load (a =1.0), the ratio between the deviations of the standard MC and the inverse convolution method (MC-IC) is about 30 in both cases, and between MC and MC-ICSA, about 190 in case 1 and 180 in case 2, corresonding to a decrease of 900 to in the required samle sizes. Also, we can note here that with the inverse convolution method the estimation of the variance of the estimates is guaranteed to be reliable. In rare event simulation (which is not the main interest here), one roblem is that one can get results that aear to be very accurate udging from the estimated variance, but the results can, in reality, be far from the correct value. This can haen, e.g. when using a single heavily twisted IS distribution, and the reason is that the likelihood ratio w( ) can have a huge value at some oint in the state sace, but under the twisted distribution these oints are very rare and we never encounter them during the course of a simulation run. Hence, the estimates, esecially for the variance or other higher moments, can be heavily underestimated, as has been rigorously shown in [10]. With the inverse convolution method, however, the estimation is always reliable, since the observed values of the samles are bounded within the interval [0, 1]. Thus, the roblem of the occurrence of events with a very small robability under the IS distribution but having a significant contribution to the estimate does not occur. A second examle used was a network having a larger number of channels, thus generating a somewhat larger state sace, big enough for not being solvable with the exact algorithm [4]. In this case, we have the same network toology as in the first examle, but use I =50channels. The link caacities are also increased: the leaf link caacities C u =30, root link caacity C J =35, and the middle link caacity C =33. All the other arameters remained the same as in the first examle. We estimated the blocking robability of a three link route from the user to the root node. The results are resented in table II. For examle, the ratio between the deviations of the standard MC and MC-ICSA is about 20 in both cases for heavy load (a = 700), and about 140 for light load (a = 400), corresonding to a decrease of 400 to in the required samle sizes. To get a feel of what kind of reductions in the run time one can obtain with the inverse convolution method, we also rovided the actual CPU-time used by our simulator imlemented in Matlab. Note that the classic MC algorithm could be imlemented using matrix oerations, which in this case increases the seed difference in favour of the classic MC. In site of this, we see that the time used er samle in MC-IC is only five times as long as for the classic MC. Increasing the number of samles for the classic MC even ten times (case 2), however, does not bring the relative deviation anywhere near to the one of MC-IC. We also see that the added enalty for samle allocation is small enough to ustify its use. The added cost for rearing simulation in MC-IC and MC-ICSA was small, about 80 ms even for this larger network. TABLE II The relative deviation of the estimatesp (B u), and CPU time for simulation, for the second examle. Case 1: samles, Case 2: samles. Case a / B u relative deviation/elased time MC MC-IC MC-ICSA 400 / / / / % 7.0 s 37 s 43 s 700 / / / / 10% 6.9 s 36 s 41 s 400 / / / 0.02/ % 70 s 370 s 420 s 700 / / 0.05/ % 69 s 360 s 400 s VI. Conclusions In this aer we have resented a new aroach to the roblem of estimating blocking robabilities in a multicast network by using the static Monte Carlo simulation method and imortance samling. First we observed that the estimation roblem can be decomosed into searate simler sub-roblems; estimation of blocking on each link on the route, attributing each blocking state to a single link, viz. the blocking link closest to the user on the route R u. For the solution of the sub-roblems, we resented a method which very closely aroximates the generation of samles with the ideal IS distribution, and gives a remarkable variance reduction. The idea of the method is to generate samles directly into the set of blocking states of a given link in a system, where all the other links are assumed to have an infinite caacity. This //$ IEEE 438 IEEE INFOCOM 20

8 is achieved by the inverse convolution method resented in the aer. This set of blocking states of course extends beyond the allowed state sace of the system, and may generate blocking in other links, as well. Then, simulation is essentially only needed to determine which art of this set is actually inside the allowed state sace, and which art of this set adds to the attributed blocking of the link. The inverse convolution method can easily be modified to cover the case studied by Nyberg et al. [4], where the multicast network model was extended to include indeendent background traffic on the links. This extension includes the from a ractical oint of view imortant case where the multicast tree is a art of a larger network carrying also unicast traffic. Acknowledgement This research has been funded by the Academy of Finland. The work of the first author has also been financially suorted by the Nokia Foundation. References [1] J. Karvo, J. Virtamo, S. Aalto, and O. Martikainen, Blocking of dynamic multicast connections in a single link, in Proceedings of Broadband Communications 98, Aril 1998, [2] K. Boussetta and A. L. Belyot, Multirate resource sharing for unicast and multicast connections, in Proceedings of Broadband Communications 99, November [3] J. Karvo, J. Virtamo, S. Aalto, and O. Martikainen, Blocking of dynamic multicast connections, to aear in Telecommunication Systems. [4] E. Nyberg, J. Virtamo, and S. Aalto, An exact algorithm for calculating blocking robabilities in multicast networks, in Proceedings of Networking 2000, Paris, May 2000, [5] P. E. Lassila and J. T. Virtamo, Efficient imortance samling for monte carlo simulation of loss systems, in Proceedings of the ITC-16. June 1999, , Elsevier. [6] M. Mandes, Fast simulation of blocking robabilities in loss networks, Euroean Journal of Oerations Research, vol. 1, , [7] K. W. Ross, Multiservice Loss Models for Broadband Telecommunication Networks, Sringer-Verlag, [8] P. Lassila and J. Virtamo, Nearly otimal imortance samling for monte carlo simulation of loss systems, to aear in: ACM Transactions on Modeling and Comuter Simulation. [9] E. Nyberg, J. Virtamo, and S. Aalto, An exact algorithm for calculating blocking robabilities in multicast networks, submitted for ublication. [10] J. S. Sadowsky, On the otimality and stability of exonential twisting in monte carlo estimation, IEEE Transactions on Information Theory, vol. 39, , //$ IEEE 439 IEEE INFOCOM 20