framework based on probabilistic graphical models to tackle this task. Specifically, we propose to use factor graphs to model the stationary distribution of ...
Calculating Blocking Probabilities for Loss Networks Based on Probabilistic Graphical Models Jian Ni and Sekhar Tatikonda Department of Electrical Engineering, Yale University, New Haven, CT 06520, USA Email:{jian.ni, sekhar.tatikonda}@yale.edu Abstract— Loss networks are a class of resource-sharing models which provide a powerful tool to the analysis and design of many communications and networking systems. For most loss networks of practical interest calculating the exact blocking probabilities is a difficult task. In this paper we present a new framework based on probabilistic graphical models to tackle this task. Specifically, we propose to use factor graphs to model the stationary distribution of product-form loss networks. We also propose to use the sum-product algorithm to compute the marginal distributions and the blocking probabilities of all call classes. Through extensive numerical experiments we show that the sum-product algorithm returns very accurate blocking probabilities and greatly outperforms the reduced load approximation for both single-service and multiservice loss networks with a variety of topologies. In addition, the sum-product algorithm converges very fast and can be implemented in a distributed way.
I. I NTRODUCTION Loss networks are a class of resource-sharing models which provide a powerful tool to the analysis and design of many communications and networking systems, including telephone networks, circuit-switched networks, ATM networks, mobile cellular systems, wavelength-division multiplexing optical networks, and many other computer and social systems [4], [9]. For a loss network, knowing the blocking probabilities is an essential component for evaluating the system performance (throughput, revenue rate, etc.), designing high-performance call admission control (CAC) policies and routing schemes, and implementing efficient pricing mechanisms. However, for most loss networks of practical interest, efficiently calculating the blocking probabilities is a difficult task. We introduce a general model of loss networks as follows. The network has J links, labelled 1, 2, ..., J. Link j has Cj units of resources. Let K be the set of call classes, with |K| = K. A class-k call requires Ajk units of resources on link j, where Ajk is a nonnegative integer. Ajk = 0 implies that class-k calls do not use link j. We call a loss network single-service if Ajk takes value 0 or 1 only, and multiservice if Ajk can take different positive integer values. Let Jk = {j : Ajk > 0} be the set of links (the route) used by class-k calls, with |Jk | = Jk . Let Kj = {k : Ajk > 0} be the set of call classes that use link j, with |Kj | = Kj . Class-k calls arrive according to a Poisson process with rate λk independent of the arrival processes of calls of other classes. A class-k call is blocked and lost if, on any link j ∈ Jk , there are fewer than Ajk units of free resources. Otherwise the call is accepted and simultaneous uses Ajk units of resources from every link j ∈ Jk for an arbitrarily
distributed service time with mean 1/µk . This service time is independent of the arrival and service times of other calls. Let n = (nk : k ∈ K) denote the state of the network, where nk is the number of class-k calls that are currently in the network. Let C = (C1 , ..., CJ ) be the vector of link capacities and A = (Ajk : j = 1, ..., J; k ∈ K) be the routing matrix. Hence the set of all possible states is denoted by: ∆
S(A, C) = {n ∈ ZK + : An ≤ C}.
(1)
It is well known [4], [9] that the steady-state probability of the network being in state n ∈ S(A, C) is given by: Y ρn k k π(n) = G(A, C)−1 , (2) nk ! k∈K
where ρk = λk /µk is the offered load of class-k calls, and G(A, C) is the normalization constant or partition function: X Y ρn k ∆ k G(A, C) = . (3) nk ! n∈S(A,C) k∈K
Equation (2) is often referred to as the product-form distribution of a loss network. The insensitivity property [1], [3], [9] states that the distribution holds for general service time distributions with finite means. Once the normalization constant and the steady-state probabilities are known, performance metrics of interest, such as the blocking probabilities, can be directly evaluated. The explicit and simple form of the stationary distribution seems to provide a complete solution. However, for most loss networks of practical interest, calculating the normalization constant is difficult, because it requires summing over all the states in S(A, C), the cardinality of which grows exponentially with the number of call classes, links, and the capacities of the links. In fact [7] has shown that exactly calculating the blocking probabilities of a loss networks is a #P -complete problem. There are several techniques proposed for approximating the normalization constant and the blocking probabilities. The Erlang fixed point approximation (EFPA) and more generally the reduced load approximation (RLA) assume that blocking in the network occurs independently from link to link, and the offered load of a call class on a link is reduced by blocking on other links [2], [4], [9], [12]. This leads to a set of fixed point equations, the solution of which provides the approximate link blocking probabilities. Monte Carlo summation is another technique for estimating the normalization constant and the blocking probabilities for loss networks with arbitrary topology, but convergence to the exact blocking probabilities is typically slow [8]. In addition,
for large networks it may face numerical overflow problem when estimating the normalization constant. Most importantly, none of the above techniques fully takes advantage of the spatial properties of a loss network. One work does so is [13]. There the authors examined the connection between loss networks and Markov random field theory. However, their model is quite limited and they do not though examine the use of efficient message-passing algorithms. In this paper we present a new framework based on probabilistic graphical models that fully takes advantages of the spatial properties of a loss network and message-passing algorithms. This approach can be applied to other product-form stochastic networks including multiclass queueing networks. We organize the paper as follows. In Section II we use factor graphs to model the stationary distribution of product-form loss networks. We also introduce the sum-product algorithm for computing the marginal distributions and the blocking probabilities of all call classes. In Section III we evaluate the performance of the sum-product algorithm and the reduced load approximation for both single-service and multiservice loss networks with a variety of topologies. The paper is concluded in Section IV. II. FACTOR G RAPHS AND T HE S UM -P RODUCT A LGORITHM In this paper we will use factor graphs to model the stationary distribution of product-form loss networks. Factor graphs are a class of probabilistic graphical models that have recently attracted intense research interests in electrical engineering and computer science. A wide variety of algorithms that have been developed in artificial intelligence, signal processing, and digital communications, including the forward/backward algorithm, the Viterbi algorithm, the iterative turbo decoding algorithm, Pearl’s belief propagation algorithm for Bayesian networks, the Kalman filter, etc., can be derived as specific instances of a message-passing algorithm (the sum-product algorithm) operating on a factor graph [5]. Let f (x1 , ..., xn ) be a multivariate function that can be factorized as the product of a set of local functions: f (x1 , ..., xn ) =
M Y
fm (xCm ),
(4)
m=1
where Cm ⊆ {1, 2, ..., n} indexes those variables that are arguments of the local function fm . D EFINITION : A factor graph is a bipartite graph that expresses the factorization structure of a multivariate function as defined in (4): 1) it has a variable node for each variable xi ; 2) it has a factor node for each local function fm ; 3) there is an edge connecting variable node xi to factor node fm if and only if xi is an argument of fm . We can use a factor graph to model the stationary distribution of a product-form loss network. To see this note that (2) can be written as: π(n) ∝
Y k∈K
fk (nk )
J Y
gj (nKj ).
(5)
j=1
The factor graph has a variable node for each random variable nk . For each call class k ∈ K, the factor graph has a factor node fk with the local function fk (nk ) = ρnk k /nk !,
which expresses the (unnormalized) marginal distribution of nk if there is no link capacity constraint. For each link j = 1, ..., J, the factor graph has a factor node gj with the local function P 1 if k∈Kj Ajk nk ≤ Cj , gj (nKj ) = 0 otherwise, which expresses the capacity constraint of link j. The factor graph has an edge connecting variable node nk to a factor node if and only if nk is an argument of the local function associated with that factor node. Once the factor graph is constructed we can apply standard message-passing algorithms to compute the marginal distribution π(nk ) of nk . Here we introduce the sum-product algorithm [5]. The sum-product algorithm employs two types of messages: one that flows from a variable node to a factor node; and one that flows from a factor node to a variable node. Let νnk →f denote the message sent from variable node nk to factor node f , let µf →nk denote the message sent from factor node f to variable node nk . The message computations performed by the sum-product algorithm can be expressed as follows: (SP1) variable node to factor node: Y νnk →f (nk ) = µf 0 →nk (nk ), f 0 ∈N (nk )\f
(SP2) factor node to variable node: X Y µf →nk (nk ) = {f (N (f )) N (f )\nk
νnk0 →f (nk0 )}
nk0 ∈N (f )\nk
where f (N (f )) is the local function associated with factor node f , and N represent those variable/factor nodes that are neighbors of the node in question. Cj Let Nk = minj∈Jk b Ajk c be the maximum number of class-k calls that can be in the network. Note that the message sent from or sent to variable node nk is a vector of Nk + 1 entries, which can be viewed as the ‘belief’ of the marginal distribution of nk held by the sender at the time when it sends out the message. Messages are passed (beliefs are propagated) among the nodes and are updated according to (SP1) and (SP2), so the sum-product algorithm is also called a message-passing algorithm or belief propagation algorithm. In the case when the factor graph is a tree, the sum-product algorithm operates according to the following protocol: Message-Passing Protocol for Tree Factor Graphs. A node can send a message to a neighboring node when and only when it has received messages from all of its other neighbors. It is easy to check that for a tree factor graph under this protocol the sum-product algorithm will terminate once two messages have been passed over every edge, one in each direction. Moreover, when the sum-product algorithm terminates, at variable node nk , the product of all incoming messages, suitably normalized, will be equal to the marginal distribution π(nk ) [5]: Y π(nk ) ∝ µf →nk (nk ). (6) f ∈N (nk )
Once the marginal distribution of nk is known, we can calculate the blocking probability Bk of class-k calls via
Little’s Theorem, i.e., E[nk ] = ρk (1 − Bk ). In the case when the factor graph has cycles (loopy factor graphs), there are two main approaches to deal with it. One approach involves constructing a junction tree [6]. The other is to perform the sum-product algorithm directly on the graph with cycles [10]. We adopt the second approach in this paper, using the following message-passing schedule: Message-Passing Schedule for Loopy Factor Graphs. Messages are passed among the nodes iteratively. During each iteration, in the first phase, each variable node sends messages to all its neighbors (factor nodes); and in the second phase, each factor node sends messages to all its neighbors (variable nodes). The algorithm terminates when the messages converge. When the messages converge, we can use (6) to approximate the marginal distributions of the variable nodes, and then calculate the approximate blocking probabilities of all classes via Little’s Theorem. III. N UMERICAL E XPERIMENTS AND A NALYSIS In this section we will evaluate the performance of the sum-product algorithm for loss networks and compare it with reduced load approximation (RLA). For all numerical experiments, let Bk∗ be the exact blocking probability of class¯ ∗ = PK B ∗ /K be the average blocking k calls. Let B k=1 k probability of the network, which can be used as an indication of the congestion level of the network. Let Bks be the exact (for tree factor graphs) or approximate (for loopy factor graphs) blocking probability of classk calls returned by the sum-product algorithm. Let Bkr be the approximate blocking probability returned by RLA: for single-service loss networks, EFPA is applied to calculate Bkr ; for multiservice loss networks, the knapsack approximation is applied to calculate Bkr . A. Reduced Load Approximation for Loss Networks The Erlang fixed point approximation (EFPA) is a special case of RLA in which blocking on a link is modelled using the Erlang loss model. Let Lj denote the approximate probability that “there is no free resource on link j”. Assume these events are independent from link to link. EFPA induces a set of fixed point equations as follows: Lj = E(Cj , ηj ) : j = 1, 2, ..., J, (7) X Y where ηj = (1 − Lj )−1 ρk Ajk (1 − Li )Aik k∈Kj
i∈Jk
is the “reduced load” of all classes offered to link j and E is C . the Erlang’s loss formula E(C, η) = PCη /C! n n=0 η /n! One can use repeated substitutions to find a fixed point solution of the equations induced by EFPA [9], [12]. The approximate blocking probability of class-k calls then is given Q by Bkr = 1 − j∈Jk (1 − Lj )Ajk . For single-service loss networks EFPA is quite accurate for many topologies of practical interest [12], but for multiservice loss networks it may produce very inaccurate results. For multiservice loss networks we introduce another RLA approach that yields better results than EFPA: the knapsack approximation, in which blocking on a link is modelled using the stochastic knapsack model [2].
Let Ljk denote the approximate probability that “there are fewer than Ajk units of free resources on link j.” Assume these events are independent from link to link. The knapsack approximation induces a set of fixed point equations: Ljk = Kjk (Cj , ηjl : l ∈ Kj ) : k ∈ Kj , j = 1, 2, ..., J, (8) Q where ηjl = ρl i∈Jl \j (1 − Lil ) is the “reduced load” of class-l calls offered to link j. The function Kjk , for k ∈ Kj , can be computed using the Kaufman’s recursive algorithm [2], [3], with a computational complexity O(Cj |Kj |). The authors of [2] suggested to use repeated substitutions to find a fixed point solution of the set of equations. If repeated substitutions converge, we can approximate the blocking probQ ability of class-k calls by Bkr = 1 − j∈Jk (1 − Ljk ). B. Loss Networks with Tree Factor Graphs For loss networks with tree factor graphs, the sum-product algorithm is exact, so Bk∗ = Bks . We use the exact blocking probabilities returned by the sum-product algorithm to evaluate the performance of RLA. For each class k, let rk = |Bkr − Bk∗ |/|Bk∗ | be the relative error of Bkr to the exact blocking probability. Let rmax = maxk rk be the maximum relative error (among all call P classes) and rave = k rk /K be the average relative error of RLA. Let Mr be the number of iterations required by repeated substitutions to solve RLA equations. 1) Linear Loss Networks: Consider a linear network consisting of J links which are arranged linearly. Single-link calls of class k = j, j = 1, 2, ..., J, arrive with offered load ρk and each requires bk units of resources from link j. Two-link calls of class k = (j, j + 1), j = 1, 2, ..., J − 1, arrive with offered load ρk and each requires bk units of resources from links j and j + 1. Therefore there are totally K = 2J − 1 call classes. It is easy to check that the associated factor graph is a tree. We did numerical experiments for a linear network with 20 links and 39 call classes. The link capacity Cj is chosen at random uniformly among integers in [80,120]. We considered both single-service cases in which bk = 1 for all k, and multiservice cases in which bk is chosen at random uniformly among integers in [1,5]. We also consider different cases under light, moderate, and heavy offered loads respectively. The results are shown in Table I. We found that for all cases the average relative error of RLA is around 10%, which implies that RLA returns fairly accurate blocking probabilities for most call classes; but the maximum relative error can be very large, which implies that RLA may return very inaccurate blocking probabilities for specific call classes. We did similar experiments for other linear networks (J = 40, 100, etc.) and we observed the same pattern of the results. 2) Tree Access Loss Networks: A tree access network [11] is defined as follows: 1) the network consists of J access links, labelled 1, 2, ..., J, and one common link labelled 0; 2) there are K = J classes of calls: class-k (1 ≤ k ≤ J) calls arrive with offered load ρk and each requires bk units of resources both on the kth access link and the common link. The common link has capacity C0 and the kth access link has capacity Ck . We did numerical experiments for a tree access network with 20 access links and 20 call classes. The results are shown in Table II. In Table II(a), when the capacity of the common link C0 = 1000 is much less than the aggregate capacity of all access
TABLE I N UMERICAL E XPERIMENTS F OR L INEAR L OSS N ETWORKS
bk
ρk bk
J = 20, K = 39, Cj =U[80,120] ¯∗ B RLA sum-product
TABLE III N UMERICAL E XPERIMENTS F OR R ING L OSS N ETWORKS J = 20, K = 40, Cj =U[80,120] bk
Mr
rave (%)1
rmax (%)
ρk bk
¯∗ B
Sum-Product
RLA
simulation
Ms
save
smax
Mr
rave
rmax
1
20
0.000320
4
2.1
28.9
1
30
0.041187
10
0.8
4.6
14
9.5
48.6
1
30
0.041138
18
8.6
84.5
1
40
0.177694
17
0.1
0.5
28
1.8
9.3
1
40
0.205344
30
9.1
233.6
1
50
0.307155
20
0.1
0.4
36
0.3
0.7
U[1,5]
20
0.008360
9
17.0
155.2
U[1,5]
30
0.097344
12
0.3
1.5
17
6.8
60.5
U[1,5]
30
0.097362
16
11.2
192.3
U[1,5]
40
0.234391
16
0.1
0.4
27
1.9
17.8
U[1,5]
40
0.201130
27
4.2
85.3
U[1,5]
50
0.352575
18
0.1
0.4
32
0.7
6.6
1. The relative errors here and also in other tables are expressed in percentage.
TABLE II N UMERICAL E XPERIMENTS F OR T REE ACCESS L OSS N ETWORKS (a) C0 =1000 bk
J = 20, K = 20, Cj =U[80,120] for 1 ≤ j ≤ J ¯∗ ρk bk B RLA sum-product
Mr
rave
rmax 0.001
1
60
0.170613
6
0.0002
1
80
0.376033
7
0.0001
0.001
1
100
0.500498
8
0.00004
0.0003
U[1,5]
60
0.179723
11
0.09
0.3
U[1,5]
80
0.379010
13
0.06
0.1
U[1,5]
100
0.501545
15
0.05
0.3
(b) C0 =2000 J = 20, K = 20, Cj =U[80,120] for 1 ≤ j ≤ J ¯∗ bk ρk bk B RLA sum-product
Mr
rave
rmax
1
80
0.022045
2
37.2
99.8
1
100
0.126191
6
23.5
89.1
1
120
0.246802
16
13.9
59.2
U[1,5]
80
0.084962
4
15.6
91.9
U[1,5]
100
0.155701
8
14.6
69.3
U[1,5]
120
0.256524
21
12.4
61.9
links (2000 on average), which means that the common link is the bottleneck of the network, we found that RLA returns very accurate blocking probabilities: the maximum relative error is less than 1%! In Table II(b), when the capacity of the common link C0 = 2000 is close to the aggregate capacity of all access links, we found that RLA produces significant errors. We did similar experiments for other tree access networks (J = 10, 40, etc.) and we observed the same pattern of the results. C. Loss Networks with Loopy Factor Graphs For loss networks with loopy factor graphs the sum-product algorithm is not exact. In order to evaluate its performance and the performance of RLA, we use discrete-event simulation to estimate the exact blocking probabilities. Let sk = |Bks − Bk∗ |/|Bk∗ | be the relative error of Bks returned by the sum-product algorithm to the exact blocking
probability. Let smax maxk sk be the maximum relative P = s s error and ave = k k /K be the average relative error of the sum-product algorithm. Let Ms be the number of iterations required by the sum-product algorithm to converge. 1) Ring Loss Networks: Consider a ring network consisting of J links which are arranged as a ring. Single-link calls of class k = j, j = 1, 2, ..., J, arrive with offered load ρk and each requires bk units of resources from link j. Two-link calls of class k = (j, (j mod J) + 1), j = 1, 2, ..., J, arrive with offered load ρk and each requires bk units of resources from links j and (j mod J) + 1. We did numerical experiments for a ring network with 20 links and 40 call classes. We considered both single-service cases and multiservice cases, and under light, moderate, and heavy offered loads respectively. The results are shown in Table III. We found that for all cases the performance of the sum-product algorithm is consistently superior and always better than RLA: with an average relative error less than 1% and a very small maximum relative error. RLA also returns fairly accurate blocking probabilities, but not as accurate as the sum-product algorithm: with an average relative error around 10% and sometimes a high maximum relative error. 2) Star Loss Networks: Consider a star network consisting of J leaf nodes and one central node. Each leaf node is connected to the central node so there are totally J links. Calls of class k = (i, j), i 6= j, arrive with offered load ρk and each requires bk units of resources from each of the two links connecting the leaf nodes i and j. Hence there are totally K = J(J − 1)/2 call classes. We did numerical experiments for a star network with 10 links and 45 call classes. The results are shown in Table IV, which are very similar as those of the ring network: for all cases the performance of the sum-product algorithm is consistently superior and always better than RLA. D. Random Loss Networks Finally we consider random loss networks with J links and K call classes. For each call class k with probability p it requires bk units of resources from link j, j = 1, 2, ..., J. We choose p such that the average number of links used by class-k ¯ calls is pJ = J. We did numerical experiments for a random network with 100 links, 50 call classes, J¯ = 6. The results are shown in Table V, which follow the same pattern as for the previous network topologies: the performance of the sum-product algorithm is consistently superior and always better than RLA.
TABLE IV N UMERICAL E XPERIMENTS F OR S TAR L OSS N ETWORKS J = 10, K = 45, Cj =U[80,120] bk
ρk bk
¯∗ B simulation
Sum-Product
RLA
Ms
save
smax
Mr
rave
rmax
1
8
0.016374
9
2.9
28.6
9
3.9
57.1
1
10
0.080058
16
1.0
8.7
17
2.2
16.2
1
12
0.184972
28
0.6
4.5
32
0.9
9.1
U[1,5]
8
0.057061
13
0.5
4.7
13
2.5
10.3
U[1,5]
10
0.142942
19
0.4
2.5
21
1.6
10.1
U[1,5]
12
0.233794
28
0.4
2.3
30
0.9
5.0
TABLE V N UMERICAL E XPERIMENTS F OR R ANDOM L OSS N ETWORKS J = 100, K = 50, J¯ = 6, Cj =U[80,120] bk
ρk bk
¯∗ B simulation
Sum-Product
RLA
Ms
save
smax
Mr
rave
rmax
1
10
0.017825
8
1.3
12.4
8
3.6
22.9
1
20
0.228643
34
1.1
30.7
52
9.3
220.3
1
30
0.414611
74
0.6
8.7
537
6.5
147.5
U[1,5]
10
0.015772
10
5.3
49.1
10
21.6
155.2
U[1,5]
20
0.179833
39
1.6
17.0
72
7.7
58.4
U[1,5]
30
0.370097
128
0.4
2.7
105
82.4
100.0
There also exists a case for which repeated substitutions for solving RLA equations do not converge after a predetermined threshold (case where Mr = 105 ) and for that case RLA returns very inaccurate blocking probabilities for all call classes. E. Computational Complexity Analysis We observed that the sum-product algorithm converges very fast, for all experiments it terminates instantaneously or within a few seconds, running on a PC with Pentium III 933MHz CPU. We now analyze the computational complexity of the sum-product algorithm for loopy factor graphs under the iterative message-passing schedule. During each iteration, in the first phase, every variable node sends messages to all its neighbors. To compute the message νnk →gj for each j ∈ Jk , it requires Nk + 1 multiplicaCj tions, with Nk = minj∈Jk b Ajk c. Hence the computational P complexity in this phase is O( k∈K Jk Nk ). In the second phase, every factor node sends messages to all its neighbors. To compute the message µgj →nk for each k ∈ Kj , we have applied the convolution algorithm introduced in [11], with a computational complexity of O(Kj2 Cj2 ). Hence the PJ computation complexity in this phase is O( j=1 Kj2 Cj2 ). Therefore, the total computational complexity in each iteration ¯ 2 C¯ 2 ) where K ¯ is the of the sum-product algorithm is O(J K average number of classes using a link and C¯ is the average link capacity. IV. C ONCLUSIONS In this paper we presented a new framework based on probabilistic graphical models to calculate the blocking probabilities of loss networks. Specifically, we proposed to use
factor graphs to model the product-form stationary distribution of a loss network, which fully takes advantage of the spatial properties of the loss network. We also introduced the sumproduct algorithm for computing the marginal distributions and the blocking probabilities of all call classes, which is exact for loss networks with tree factor graphs. Through extensive numerical experiments we showed that the sumproduct algorithm returns very accurate blocking probabilities and always outperforms the reduced load approximation for both single-service and multiservice loss networks with a variety of topologies. In addition, the sum-product algorithm is computationally efficient and converges very fast. Another merit of the sum-product algorithm is, noting that the message computations in (SP1) and (SP2) are purely local so that it can be implemented in a distributed way. The factor graph approach can be applied to analyze other product-form stochastic networks including multiclass queueing networks. Currently we are applying this approach to calculate the normalization constant and performance metrics of closed queueing networks. In the future, we will study the connection between the network topology and the performance of the sum-product algorithm for general loss networks (esp., loss networks with loopy factor graphs), to provide certain theoretical bounds of the accuracy and the convergence rate of the algorithm. We will also apply this powerful tool to the design of highperformance CAC policies, routing schemes and pricing mechanisms in loss networks. R EFERENCES [1] D. Y. Burman, J. P. Lehoczky, Y. Lim, “Insensitivity of Blocking Probabilities in a Circuit-Switching Network,” Journal of Applied Probability, vol. 21, 1984, pp. 850-859. [2] S. P. Chung, K. W. Ross, “Reduced Load Approximation for Multirate Loss Networks,” IEEE Transactions on Communications, vol. 41, no. 8, Aug. 1993, pp. 1222-1231. [3] J. S. Kaufman, “Blocking in a Shared Resource Enviroment,” IEEE Transactions on Communications, vol. 29, no. 10, Oct. 1981, pp. 14741481. [4] F. Kelly, “Loss Networks,” The Annals of Appleid Probability, vol. 1, no. 3, 1991, pp. 319-378. [5] F. R. Kschischang, B. J. Frey, H.-A. Loeliger, “Factor Graphs and the Sum-Product Algorithm,” IEEE Transactions on Information Theory, vol. 47, no. 2, Feb. 2001, pp. 498-519. [6] S. Lauritzen, Graphical Models, Clarendon Press, Oxford Statistical Science Series, 1996. [7] G. Louth, M. Mitzenmacher, F. Kelly, “Computational Complexity of Loss Networks,” Theoretical Computer Science, vol. 125, 1994, pp. 4559. [8] K. W. Ross and J. Wang, “Monte Carlo Summation Applied to ProductForm Loss Networks,” Probability in the Engineering and Informational Sciences, vol. 6, 1992, pp. 323-348. [9] K. W. Ross, Multiservice Loss Models for Broadband Telecommunication Networks, Springer, 1995. [10] S. Tatikonda and M. Jordan, “Loopy Belief Propagation and Gibbs Measures,” Uncertainty in Artificial Intelligence (UAI), Proceedings of the Eighteenth Conference, 2002. [11] D. H. K. Tsang and K. W. Ross, “Algorithms to Determine Exact Blocking Probabilities for Multirate Tree Networks,” IEEE Transactions on Communications, vol. 38, no. 8, Aug. 1990, pp. 1266-1271. [12] W. Whitt, “Blocking When Service is Required from Serveral Facilities Simultaneously,” AT&T Technical Journal, vol. 64, 1985, pp. 1807-1856. [13] S. Zachary and I. Ziedins, “Loss Networks and Markov Random Fields,” Journal of Applied Probability, vol. 36, no. 2, 1999, pp. 403-414.
