Performance Analysis of Queueing Networks with Blocking

Transcription

1 Performance Analysis of Queueing Networks with Blocking Simonetta Balsamo Dipartimento di Informatica, Università Ca Foscari di Venezia, Venice, Italy Abstract Queueing network models have been widely applied for performance modeling and analysis of computer networks. Queueing network models with finite capacity queues and blocking (QNB) allow representing and analyzing systems with finite resources and population constraints. Different protocols can be defined to deal with finite capacity resources and they can be modeled in queueing networks with blocking by various blocking types or mechanisms. Modeling heterogeneous networks having different blocking protocols leads to heterogeneous QNB where the service centers may have different blocking types. QNB are difficult to analyze, except for the special class of product-form networks. Most of the analytical methods proposed in literature provide an approximate solution with a limited computational cost. This tutorial introduces queueing network models with finite capacity queues and various types of blocking. We discuss the main solution techniques for their exact and approximate analysis to derive network performance indices. We present and compare the solution methods for open and closed networks to identify the criteria for the appropriate selection of a solution method. We provide some application examples of this class of models to computer and communication networks. 1. Introduction Queueing network models have been widely applied for performance modeling, analysis and prediction of various systems, including communication and computer networks. They allow representing resource contention by a set of customers. The servers of the queueing network represent the resources and the customers that queue to obtain service represents the jobs, tasks, packets or messages that are the system workload. System performance analysis based on queueing networks consists of the evaluation of a set of figures of merit such as some average performance indices that include system throughput, resource utilization, customer s mean response time or delay. The analysis of queueing networks is usually based on the definition of the underlying stochastic process, specifically a continuous time Markov chain. Some efficient solution algorithms have been defined for the special class of product-form queueing networks [Kl76, Ka92, L83, T01, BCMP75]. Some important characteristics of resource sharing systems such as computer and communication networks that affect system performance are the finite buffer space of the queues and possible constraints on the number of customers in the network or in a subnetwork. Queueing network models with finite capacity queues and blocking (QNB) allow representing and analyzing systems with finite resources and population constraints [BDO01, P94]. Different protocols can be defined to deal with finite capacity resources and they can be modeled in queueing networks with blocking by various blocking types or mechanisms. Modeling heterogeneous networks having different blocking protocols leads to heterogeneous QNB where the service centers may have different blocking types. T6/1

2 QNB are difficult to analyze, and in general their steady state queue length distributions could not be shown to have product form solutions, except for some special cases [BD94, BDO01]. Hence, most of the solution methods are approximations, simulation, and numerical techniques. Most of the analytical methods proposed in literature provide an approximate solution with a limited computational cost [BDO01, P94] This paper introduces QNB, the main solution techniques to derive performance indices and some useful properties. Section 2 describes the class of queueing network models with finite capacity queues and various types of blocking. In Section 3 we discuss the main solution techniques for their exact analysis of QNB to derive network performance indices. Sections 4 and 5 deal with the approximate analysis of QNB for open and close network model, respectively. We compare the various analysis techniques to identify the criteria for the appropriate selection of a solution method. Section 6 presents some application examples of QNB to model heterogeneous computer and communication networks. Conclusions and open problems are given is Section Queueing Networks with blocking A queueing network models represent resource sharing system where a set of customers ask for service to a set of resources. In a queueing network model of computer and communication networks customers can model packets, messages, job and tasks and the time required for the service is usually represented by a random variable. Consider a queueing network formed by M service centers or nodes. A queueing network can be either closed or open. In a closed network there is a constant number of customers N that circulate in the network. For open networks we define an external arrival process at each node i, 1 i M. Let l denote the total arrival rate at the network. We usually assume a Poisson arrival process. Let p 0i denote the probability that an external arrival tries to enter node i. Then the arrival process at node i is Poisson with parameter lp 0i. The customers in the queueing network move among the service centers according to routing probabilities. Let P=[p ij ] denote the routing probability matrix, where p ij denotes the probability that a customers leaving node i immediately moves to node j, 1 i,j M. The traffic balance equations are defined as the following linear system: M x i = Â x j p ji + lp 0i (1) j=1 For open networks with an irreducible routing matrix P, this system has a unique solution x=[x 1,, x M ] that represent the throughput of the nodes. For closed networks the traffic equations system (1) does not have a unique solution and x i represents the average visit ratio at node i, 1 i M. For each service center i we define the number of servers, the service time distribution, the queue capacity and the service discipline. According to Kendall s notation a service center is denoted by A/B/s/c/D where A and B are respectively the customer interarrival time and the service time distribution, integer numbers s and c denote respectively the number of identical servers and the queue capacity, and D the service discipline. Examples of interarrival and service time distributions are exponential (M), phase-type formed by exponential stages (PH), general (G) and generalized exponential (GE). Let µ i denote the service rate of node i, i.e., 1/µ i is the mean service time. Let K i denotes the number of identical servers at node i. Let µ i f i (n i, Ki) denote the service rate at node i that depends on the number of customers in its queue, n i 0, where f i is an arbitrary non-negative function with f i (0, Ki)=0. For a single server node we denote this function as f i (n i ). Different types of customers can represent various behaviors of the jobs in the network in terms of T6/2

3 routing probability and service times. Hereafter, for the sake of simplicity, we consider single class queuing networks. Let B i denote the maximum number of customers admitted at node i that is in the queue and the servers (B i = c+s), 1 i M. Let n i denote the number of customers in node i. Then it satisfies the constraint 0 n i B i. When the total number of customers reaches the node finite capacity (n i = B i ) the node is said to be full and blocking arises. 2.1 Definition of blocking mechanisms Various protocols have been defined to represent different system behaviors with QNB. When a customer attempts to enter a queue that has reached its maximum capacity, i.e., queue is full, then the arriving customer is blocked. A blocking mechanism describes the behavior of the arriving customer. We shall now introduce the three main blocking mechanisms or types defined in the literature. They are called Blocking After Service, Blocking Before Service and Repetitive Service Blocking. Blocking After Service (BAS): when a job upon completion of its service at node i attempts to enter to a full queue (i.e., node i is blocked), it is forced to wait at node i occupying the server space, until a space becomes available at destination node j. Server i is forced to stop processing jobs that might be waiting in its queue (it is blocked) until the blocked customer can enter its destination queue. Upon departure from node i the server space becomes available for another job waiting in the queue. In this case, service at node i is resumed as soon as a departure occurs from node j (at which time, the job waiting at node i moves to its destination node immediately). If more than one node is blocked by the same destination node, then it is necessary to define the order in which blocked customers will be unblocked, as space becomes available in the blocking node. First Blocked First Unblocked discipline states that the first customer that joins the blocking queue when a departure occurs is the one that was blocked first. Blocking Before Service (BBS): a job declares its destination node j before it starts receiving service at node i. The destination of the job does not change until it completes its service and departs from the queue. If upon entering service at node i, a job finds node j full, the service at node i does not start and server i is blocked. If a destination node j becomes full prior to service completion at node i, then the service at node i is interrupted and the server is blocked. The service at node i is resumed as soon as a departure occurs from node j. It is generally assumed that the amount of service a job has received until it is blocked is lost, i.e. a new service starts when a node becomes unblocked. Two subcategories can be introduced [O90] depending on whether the service space is used to hold a customer during the time the server is blocked BBS-SO (server occupied) and BBS-SNO (server is not occupied). The latter is applicable only in network topologies in which a node with a finite capacity has only one upstream node with a finite capacity feeding it. Another variant of BBS is BBS-O (Overall Blocking Before Service) when a full node causes all its upstream nodes to be blocked, regardless of the destinations of jobs currently receiving service at each upstream node. Hereafter we consider BBS-SO blocking that is simply called BBS. Repetitive Service Blocking (RS): A job completes its service at node i and attempts to enter node j. If node j at that time is full, the job at node i starts receiving a new and independent service according to its service discipline. RS blocking is is further classified into two subcategories depending on whether a blocked job chooses a new destination node independently of its previous selection(s) (RS-RD: random destination) or it keeps trying to enter the same destination node at each service completion (RS-FD: fixed destination). A different kind of blocking mechanism called generalized blocking can be defined where the server continue processing customers in the queue even if the destination node is full. The customers that have completed service at node i but cannot be sent to the next node, continue to T6/3

4 share the buffer space of node i along with the other customers that are either waiting for service or being served upon. The customers arriving at this node when the queue is full are lost. This blocking is also called kanban blocking due to its application to modeling kanban-controlled manufacturing systems. Under this blocking mechanism a server continue the service as long as there are customers available, even if the destination node is full. Customers can be raw or completed. A raw customer has not yet been served and a completed customer has already been served by the node. The buffer at node i is divided in two portions, one for raw customers with capacity ai and the other for the finished customers with capacity bi. If the buffer for raw customers at the destination node is full, then customers wait in buffer bi at node i. The total queue capacity of node i is Bi. Hence the following relations hold: 0 ai, bi Bi and ai+bi Bi. Therefore a customer completing service at node i can be blocked and forced to wait in buffer bi if the destination buffer aj or the total capacity of node j is the maximum allowed Bj. The server of node i can be blocked if buffer bi is full and it is starved if there are no raw customers in ai. For particular values of the parameters ai and bi this blocking reduces to other blocking types. Specifically general blocking includes BBS-SO for ai=bi and bi=0, BAS for ai=bi and bi=1 for single server node or more generally, bi=ki. Other types of blocking mechanisms, allow representing network population constraints, by imposing the number of customers to be in the range [L,U], where L and U are the minimum and maximum populations admitted, respectively. Let a(n) to denote the load dependent arrival rate function and d(n) denote a non-negative departure blocking function, where n 0 is the network population, where a(n) = 0, for n U and d(n) = 0 for n L. We shall now introduce the Stop and the Recirculate blocking. In Stop Blocking, also referred to as delay blocking, the service rate at each node depends on the number of customers in the network (n), according to the function d(n). When d(n) = 0, the service at each node is stopped. Service at a node is resumed upon arrival of a new customer to the network. In Recirculate Blocking, also referred to as triggering protocol, a job upon completion of its service at node i leaves the network with probability p i0 d(n), when n is the total network population and it is forced to stay in the network with probability p i0 [1-d(n)], where p i0 is the routing probability. Consequently, a job completing the service at node i enters node j with state dependent routing probability p ij + p i0 [1-d(n)] p 0j, 1 i,j M, n Deadlock Due to the finite capacity of the queues, QNB can deadlock, depending on the blocking mechanism used to model the network and on the network topology. When two or more customers are each is waiting for the other, a deadlock occurs if a space in one of the two or more nodes never becomes available. If deadlock may occur, the problem can be solved by preventive techniques, or detection and resolve techniques. Stop and Recirculate blocking mechanisms do not cause any deadlock. A necessary condition for deadlock avoidance with BAS, BBS and RS-FD blocking types is that p ii =0, 1 i M. However, one can assume that a customer at n ode i cannot be blocked by i itself. A simple deadlock prevention technique for blocking types BAS, BBS and RS-FD is to check that the number of customers N is less than the total buffer capacities of nodes along each possible cycle in the network. In the case of RS-RD blocking, it is sufficient that routing matrix P is irreducible and N is less than the total buffer capacity of the nodes in the network. More precisely, a network with RS-RD blocking is deadlock-free if T6/4

5 N < M  B j j=1 Let c=(i 1,i 2,,i z ) denote a cycle in the network with i 1 =i z =i and routing probabilities p i jij+1 0, 1 j z. Then a network with BAS, BBS-SO, BBS-O and RS-FD blocking is deadlockfree if N < min "c  (2) Bi j ij Œ c where the minimum in equations (2) is computed over all the possible cycles in the network. Similarly, the same condition holds for a network with BBS-SNO blocking by substituting B i j with (B i j - 1) in formula (2). 2.3 Performance indices QNB performance analysis consists of the evaluation of a set of performance indices that are are defined for the entire network (global) and for each node (local). Most of them are average of performance indices and they include the utilization Ui, the throughput Xi, the mean queue length Li and the mean response time Ti, for each node i. Another local performance measure is the probability distribution of the number of customers at node i ni. Let pi(ni) denote the stationary (marginal) queue length distribution of node i, i.e., the stationary probability of having ni customers at node i in steady state, ni 0, 1 i M. Let zi denote the number of active servers at node i, which are servers that is not empty and not blocked, 0 zi min {ni, Ki}. Let xi(ni) denote the stationary queue length distribution of node i at arrival time and zi(ni,zi) denote the stationary joint distribution of variables ni and zi, i.e., the probability of ni customers and zi active servers in node i at arbitrary times, ni 0, 1 i M. A global performance index is the customer sojourn time that is the time a customer spends in the queueing network or in a part of it. The network sojourn time is the time a customer spends in the queueing network and the passage time is the time a given customer takes to traverse a path in the network, i.e. between two consecutive departures of a customer from two given nodes. The cycle time is the time between two consecutive departures of a particular customer from a node in the network. The average sojourn, passage and cycle time can be easily derived by the node mean response time and the visit ratio. For example in a closed network the average cycle time for node i is given by M  xjt j /xi and in an open network the average network sojourn time or network j=1 residence time is given by M  xjt j, where x i's are obtained by formula (1). The sojourn time j=1 distribution is an interesting and detailed performance measure of a queueing network, as well as the passage time and cycle time distribution, but few analytical results are available in the literature. For open queueing networks with finite capacity another performance index of interest is the job loss probability, denoted by Ploss, which is the probability that an arriving customer is not accepted and it is lost. Then the job loss probability can be computed from stationary queue length distribution of node i at arrival time as follows for BAS, BBS and RS mechanisms: T6/5

6 P loss =  xi (Bi) 1 i M One can derive the mean performance indices from the stationary joint distribution, which depends on the blocking type. Consider BAS, BBS and RS blocking mechanisms. When node i cannot be blocked its utilization can be simply defined as follows: min(bi,n ) min(n Ui = i,k i )  pi(n Ki i) (3) ni=1 where K i is the number of servers and B i the node i capacity. When node i can be blocked this formula gives for each blocking type the total utilization, that is the fraction of time that the node is not empty. In QNB we can define the effective utilization U i e of node i that is the probability that the servers are really providing service, i.e., they are neither empty nor blocked. This is a measure of the useful work of the whose definition depends on the blocking type as follows. For BAS and BBS blocking, including each subtype, we consider only the active servers as follows: U i e = min(b i,n ) min(n i,k i )  n i =1  z i =1 z i K i zi(ni,zi ) For RS blocking we observe that by definition the work of node i server is useful only if at the service completion time, i.e. at departure time from node i, the destination node is not full. If node i has exponential service time distribution, one can derive that: min(b i,n ) e min(k,k U i = i )  K   i p(n)pij k=1 "nœe: ni=k "j: nj <Bj where n = (n1, n2,,nm) and E is the state space of the exponential network with RS blocking. For single server service center, i.e., Ki=1, then zi=0,1 and these average performance indices can be simplified as follows: Ui =  pi(ni) ni>0 e U i =  z i(ni,1) for BAS and BBS n i >0 min(b i,n ) e U i =    p(n)pij for RS k=1"nœe: ni= k "j: n j <B j For Stop blocking, since the service in each node is stopped when total network population n reaches the minimum threshold L then one can write: U U e min(k,k i = i )  p(n) Ki  k=1 "nœe: ni= k, n>l T6/6

7 where E is the state space of the exponential network with STOP blocking, L and U are the minimum and the maximum population admitted in the network, with 0 L U, and n is the total network population when the state is n=(n 1,..., n M ), n = Â ni. 1 i M For Recirculate blocking the nodes are never blocked, because when the network population n reaches the minimum threshold L a customer leaving node i is forced to stay in the network with probability pi0 [1-d(n)] and moves to any node in the network according to the routing probabilities. Hence U i e = U i. For each blocking type node throughput is defined as follows: Â Â (4) Xi = mif i(ni,zi)zi(ni,zi ) n i 1 z i 1 For single server service center, i.e., Ki=1, then zi=0,1 this leads to: Xi = Âmif i(ni )zi(n i,1) n i >0 Note that like the utilization, in networks with blocking we can also define the effective throughput as a measure of the useful work of the node. For BAS blocking such measure is identical to the throughput defined by formula (4), because the server is active only if the node is not blocked. The same holds for BBS blocking if we assume that the service interrupted by a full destination node is resumes as soon as the node becomes unblocked. On the contrary for BBS blocking where the interrupted service is repeated and for RS blocking we define the effective throughput as a measure of the useful work of the node, given by the fraction of throughput that is not due to the service repetition because of blocking. Let X i e denote the effective throughput of node i for RS and BBS blocking, that similarly to the effective utilization can be defined as follows: min(n,b e i ) X i = Â Â mif i(n i,ki ) Â p(n)pij (5) k=1 "nœe: ni=k "j: n j <B j where n = (n1, n2,,nm) and E is the state space of the exponential network with RS blocking. In the case of constant service rate µi, the node throughput and the node effective throughput for Ki 1 servers reduces to Xi=X i e = Ui e Ki µi Xi=Ui Ki µi, X i e = Ui e Ki µi for BAS blocking for RS and BBS blocking. Mean queue length and mean response time for node i can be computed as for queueing network models with infinite capacity queues as follows: Li = Sni ni πi(ni) Ti = Li / Xi. T6/7

8 3. Exact analysis of queueing networks with blocking In this section we present the main approaches to analyze QNB with exact analytical methods. The more general approach is based on the definition of the associate Markov chain. In some special cases, QNB have a product form solution and can be analyzed by efficient solution algorithms. 3.1 Markov process analysis The analysis of queueing networks under general assumptions is based on the definition of an associated Markov chain. Let S=(S1,S2,...,SM) denote the state of the network where the state of node i, Si, depends on the type of node i and includes the number of customers ni. The network model evolution can be represented by a continuous-time ergodic Markov chain with discrete state space E and transition rate matrix Q. The stationary and transient behavior of the network can be analyzed by the underlying Markov process. Under the hypothesis of an irreducible routing matrix P, there exists the unique steady-state queue length probability distribution π = {π(s), SŒE}, which can be obtained by solving the homogeneous linear system of the global balance equations π Q = 0 (6) subject to the normalizing condition S SŒE π(s) =1 and where 0 is the all zero vector. The definition of state space E and transition rate matrix Q depends on the network definition and on the blocking type of each node [BDO01, BD94, O90, O93, P94, VD91a, VD91b]. From vector π one can derive π i, the queue length distribution of node i and the average performance indices of node i, such as throughput, average queue length and the mean response time, as shown in the previous Section. The solution of the Markov chain associated to the QNB to obtain the steady state queue length distribution requires (i) defining the network state space, (ii) the state transitions to construct the rate matrix, Q, and (iii) to solve the linear system (6). The first two steps are not always simple, since the state definition also depends on the blocking type and on the mutual dependencies due to blocking between the states of the nodes in the QNB. A detailed definition of the Markov process associated to different types of QNB is given in [BDO01]. We present an example of Markov process definition for QNB. For the sake of simplicity consider an exponential closed network with M nodes, N customers, each node i with queue capacity B i, service rates µ i and single server, and routing P. Let a i denote the minimum possible number of customers in node i, defined as follows: Ï M ai = maxì 0,N - Â Bj Ó j=1,j i Let d(ni) and b i (n i ) respectively denote an indicator function and a blocking function given by: d(ni) = Ï 0 if n i =0 Ì Ó 1 if ni>0 Ï and bi(ni)= 0 if n i =Bi Ì Ó 1 if ni<bi (7) Case 1. Markov chain of a QNB with RS-RD blocking For RS-RD blocking the node i state is simply Si=(ni) and the network space is given by T6/8

9 M E RS-RD = {(n 1, n 2,..., n M ) ai ni Bi, 1 i M,  ni =N } i=1 The state transition rate matrix Q=[q(S, S')], S, S'Œ E RS-RD, is defined as follows: q(s, S')]= d(n j ) µ j p ji bi(n i ) for S' = S + e i -e j, i j q(s, S)]= -  q( S,S") S"ŒERS-RD,S" S q(s, S')]=0 otherwise where ei denotes the M-vector with all zero components except one in i-th position. Case 2. Markov chain of a QNB with BAS blocking For BAS we have to consider the server activity and the scheduling of the nodes that are blocked by a full destination node. Let Sendi= { h phi>0, 1 h M} denotes the set of node i senders. Let t(i) be the number of servers currently blocked by node i and 0 t(i)  Kt. t Œ Send i Then for BAS blocking the node i state can be defined as follows: Si = (ni,si,mi) where ni is the number of jobs in node i, si is the number of servers of node i blocked by a full destination node and therefore containing a served job, with 0 si min{ni, Ki}, mi is the list of nodes blocked by node i defined as follows: Ï mi= Ì Ó if ni Bi [ r 1, r 2,º, r t( i) ] if ni = Bi (8) where rj is the j-th node blocked by node i, for 1 j t(i). Note that when node i is full, i.e. ni=bi, list mi is empty if no node has attempted to send a job to node i. When mi is not empty, it contains the indices of the nodes that have attempted to send a job to node i and are still blocked by node i. Moreover mi may contain more than one occurrence of the same node hœsendi if more servers of node h are blocked by node i. The "unblocking" scheduling is represented by list mi whose node indices are ordered according to the time at which they will be unblocked. For example, for the First Blocked First Unblocked (FBFU) discipline mi is a queue data structure. The complete state notation Si = (ni,si,mi) may be simplified when some components are not necessary. For example, if node i cannot be saturated (Bi N) component mi is useless because no node may be blocked waiting for room in i. If all the destination nodes of node i have infinite capacity queues, component si is useless because node i cannot be blocked. Hence the network space E BAS is defined as follows: M E BAS = {(S 1,S 2,...,S M ) S i = (n i,s i,m i ) ai ni Bi, 1 i M,  ni =N } i=1 For this blocking mechanism, differently from RS mechanism, the process state transitions do not necessarily correspond to a job transition (i.e. arrival or departure). Indeed, some process state transitions correspond to blocking and do not correspond to customer movements in the network. T6/9

10 By effect of simultaneous transitions, when a full node ends a service and in one of its upstream node there is a served job waiting, two simultaneous transitions take place: the transition from the full node and the transition from its upstream node. Multiple transitions can take place because of a chain of blocked nodes. In the following we show some examples of simultaneous transitions. Then the state transition rate matrix Q=[q(S, S')], S, S'Œ E BAS, is defined as follows: q(s, S')]= d(n j ) µ j p ji for S: s j <min{n j, K j }, m j =, n i <B i i j S' j (n' j,s' j,m' j ): n' j =n j -1, s' j =s j, m' j =m j, m' i =m i, S' i (n' i,s' i,m' i ): n' i =n i +1,s' i =s i, S' h =S h "h i,j or S: s j <min{n j, K j }, n i =B i i j S' j (n' j,s' j,m' j ):n' j =n j,s' j =s j +1,m' j =m j,m' i =Ins(m i,j),s' i (n' i,s' i,m' i ):n' i =n i,s' i =s i,s' h =S h "h i,j q(s, S')]= d(n j ) µ j p ji bi(n i ) for S: s j <min{n j, K j }, m j, S': Update(j,k), if i k then {n' i =n i +1, n' k =n k -1}  q(s, S)]= - q(s,s") S"ŒE BAS,S" S q(s, S')]=0 otherwise where Ins(mi, j) insert j in the list mi, according to the unblocking discipline, and Update(j,k) is the following algorithm to update the state of the nodes in the chain from node j backward to node k: Update (j,k): k j; while (m k ) {h=head(m k ); m' k =Cancel(m k ); s' h =s h -1; k h} and function Cancel(mi) cancels the first element of mi, function Head(mi) returns the first element of mi without changing mi. Example 1. Cyclic network with BAS blocking Consider a closed cyclic network illustrated in Figure 1.This model can be used to represent packet switching networks with fixed routing, where a physical path is set up for each user session. Consider the cyclic network of Fig.1 with M=4 nodes, N=10 customers, each node i with queue capacity B i =3 and single server. 1 2 M Figure 1. A cyclic queueing network Consider the system state ((3,1,[4]),(3,1,[1]),(3,0, ),( 1,1,[1])) where nodes 1, 2 and 4 are blocked (s 1 =s 2 =s 4 =1) with jobs waiting respectively for the full nodes 2, 3 and 1 (m 2 =[1], m 3 =[2], m 1 =[4]). Hence only node 3 can complete the service (s 3 =0) destined to node 4. When a departure occurs from node 3 and moves to node 4 then four simultaneous job transitions take place (i.e., at the same time the three waiting jobs in nodes 2, 1 and 4 move towards their destinations). The process transition is to state ((3,0, ),(3,0, ),(3,0, ),(1,0, )) where all nodes are active, si=0 and mi= "i, and ni does not change for any node i, 1 i 5. Case 3. Markov chain of a QNB with BBS blocking T6/10

11 For BBS blocking, since a job entering a server declares its destination, then node i state definition has to include the destination of job in service. Let Dest i denote the set of the destinations of node i, that is Dest i = {j p ij >0, 1 i,j M}. For the sake of simplicity, we define the following Boolean function that is true if node i can be blocked, false otherwise: Then, node i state definition is Ï Blocked(i)= true if $ jœdest i B j Ì <N Ó false if B j N " jœdest i Ï S i = (n i, NS i) Ì Ó (ni) if Blocked(i) = true if Blocked (i)= false (9) where ni is the number of jobs in node i and NSi is a state vector defined as follows. It has at most min{k i, Dest i +1} elements and its component NSi,k denotes the number of node i servers that are servicing jobs destined to node k, with kœdest i and for an open network NSi,0 denotes the number of node i servers with jobs that will leave the network. By definition the following constraint holds: Â kœdest NSi,k i» { 0} = min{ ni,ki}. NSi is not defined for nodes that cannot be blocked. Then for a closed network with N customers, the state space EBBS is defined as follows: M EBBS-SO = {(S 1,S 2,...,S M ) S i = (n i, NS i ) ai ni Bi, 1 i M, Â ni =N } i=1 In order to define the state transition matrix we have to distinguish various cases depending on whether the nodes are blocked. The state transitions rate matrix Q=[q(S, S')] is defined as follows: q(s, S')]= NSj,0 µ j for S: Blocked(j) Ÿ n j 1 S' j : n' j =n j -1 q(s, S')]= NSj,0 µ j p jh for S: Blocked(j) Ÿ n j >1 S': n' j =n j -1, NS'j,0=NSj,0-1, NS'j,h=NSj,h+1 q(s, S')]= d(n j ) µ j p ji for S: Blocked(j) Ÿ ( Blocked(i) (Blocked(i) Ÿ n i <1)) S': n' j =n j -1, n' i =n i +1 q(s, S')]= NSj,i µ j b(n i ) for S: Blocked(j) Ÿ n j 1 Ÿ ( Blocked(i) (Blocked(i) Ÿ n i 1)) S': n' j =n j -1, NS'j,i=NSj,i-1, n' i =n i +1 q(s, S')]= NSj,i µ j p jh b(n i ) for S: Blocked(j) Ÿ n j >1 Ÿ ( Blocked(i) (Blocked(i) Ÿ n i 1)) S': n' j =n j -1, NS'j,i=NSj,i-1, NS'j,h=NSj,h+1, n' i =n i +1 q(s, S')]= NSj,i µ j p ik for S: Blocked(j) Ÿ n j 1 Ÿ Blocked(i) Ÿ n i <1 S': n' j =n j -1, NS'j,i=NSj,i-1, n' i =n i +1, NS'i,k=NSi,k+1 q(s, S')]= NSj,i µ j p jh p ik for S: Blocked(j) Ÿ n j >K j Ÿ Blocked(i) Ÿ n i <K i S': n' j =n j -1, NS'j,i=NSj,i-1, NS'j,h=NSj,h+1, n' i =n i +1, NS'i,k=NSi,k+1 T6/11

12 q(s, S)]= - Â q(s,s") S"ŒE BBS,S" S q(s, S')]=0 otherwise A detailed definition of the Markov chain associated to QNB with different types of blocking, multiple servers and load dependent service rate is given in [BDO01]. Exact analysis of a QNB based on the Markov model requires its numerical solution. However, numerical solution can be applied only for small queueing networks, because the state space E of the Markov chain grows exponentially with the number of nodes and customers in the network. For open QNB the Markov chain is infinite and, unless a special regular structure of matrix Q allows deriving closed form expression of the solution π, one has to approximate the solution on a truncated state space. For closed networks the time computational complexity of liner system (6) is determined by the space state E cardinality that grows exponentially with the buffer sizes (Bi N, 1 i M) and M. Although the state space cardinality of the process can be much smaller than that the process of the same network with infinite capacity queues (which is exponential in N and M), it still remains numerically intractable as the number of model components grows. 3.2 Product form networks In some special cases, QNB have a product-form solution, under particular constraints and for various blocking types. A survey of product-form solutions of networks with blocking and equivalence properties among different blocking network models can be found in [BD94, BD01, V93]. Some efficient algorithms for some closed product-form networks with blocking have been recently defined [BC95, C95, S99]. These algorithms provide the model solution with a time computational complexity linear in the number of network components, i.e., the number of service centers and the number of customers. However, general queueing networks with blocking do not have a product-from solution and approximate analytical methods or simulation have to be applied. We shall now briefly review exact analysis for product form QNB. Approximate solution techniques for open and closed networks are presented in in Sections 4 and 5. QNB have product form solutions of the joint queue length distribution π for open or closed networks under certain constraints, depending both on the network definition and the blocking type. Product form solution can be defined as follows: p(s) = 1 G V(n ) M g i(ni) (10) i=1 where G is a normalizing constant and n is the total network population. The functions V and g i, 1 i M, are defined in terms of network parameters which include vector x defined by system (1) of traffic balance equations and service rates µ I, 1 i M, and depend on the blocking type and additional constraints. Some constraints refer to queueing networks with reversible routing matrix P, that is if the following condition holds: xi pij =xj pji, and l p0j =xj pj0 for1 i,j M, where x is defined by system (1). Figure 2 shows the cases of product form QNB for each combination of blocking type and network topology, where Fi, 1 i 8, denotes a particular product form formula according to expression (10) and there are the following additional constraints. Let B - =min1 i MBi denote the minimum node capacity of the network. T6/12

13 Blocking type BAS F1 F7 & Cond5 RS RS-RD RS-FD F1 F2 & Cond1 F3 F3 & B1< F4, F6 & Cond3 --- F2 & Cond2 1. BBS BBS-SO BBS-SNO F1 F1 & N B1+B2-2 F2 & Cond1 --- F3 & B1< --- F2 & Cond2, Stop F5 F5 F5 F5 F5 Recirculate F8 Two nodes Cyclic Central server Reversible routing Arbitrary Network topology Figure 2. Product form constraints: formulas for each blocking type and network topology. Condition 1. The non-empty condition for closed networks requires that at most one node can be empty, i.e., N B- B -. Condition 2. The strictly non-empty condition when each node can never be empty, i.e, if N>B- B -. Condition 3. It refers to a particular model of multiclass networks with parallel queues with interdependent blocking functions and service rates, and which satisfy a so-called invariant condition. See [AV90] for further details. Condition 4. It requires that each node i with finite capacity is the only destination node for each upstream node, i.e., it satisfies the following constraint: if p ji > 0 then p ji = 1, 1 j M. Condition 5. At most one node at a time can be blocked. Each product form formula F1 through F8 defines functions V and g i, 1 i M, in product form (10) and special constraints. Specifically V(n)=1 and g i (n i )= (xi / µi) n i for F1, F5, F7, F8 (11) (xi / µi) n n i b i i (l-1) fi (l) l=1 for F4 1 / ei n i for F2 T6/13

14 where xi defined by system (1), vector e =(e 1,,e M ) is derived by system e = e P', defined as follows: P'= p' ij, p' ij =µ j p ji, 1 i,j M, i j, and p' ii =1-S j i p' ji. Formula F1 applies to multiclass QNB with BCMP type nodes [BCMP75] and class independent capacities. Formula F5 applies to the same type of QNB, but with single class of customers. Formulas F7 and F8 are for multiclass QNB with nodes with exponential service time and First Come First Served discipline, and formula F7 applies for class independent capacities, while F8 requires class type fixed. Formula F4 applies to multiclass QNB with fixed type class, blocking functions dependent on class, node and chain, nodes with arbitrary service time distribution and a symmetric scheduling discipline or exponential service time distribution, which is the same for each class at the same node, when the scheduling is arbitrary. Formula F2 applies to single class networks with exponential nodes, load independent service rates. Special cases of formulas F3 and F6 are given in [AV89, AV90]. 3.3 Algorithms for product form networks Some efficient algorithms can be applied to evaluate average performance indices for QNB with product form solution (10). Since in QNB the definition of the queue length distribution and average indices depends on the blocking type, as defined in Section 2, one cannot immediately apply the known algorithms for BCMP networks, such as convolution algorithm and MVA [L83, Ka92]. The state space limitation leads to a new definition of a convolution algorithm taking into account a set of constraints on the queue lengths. Convolution The convolution algorithm for QNB can be applied to closed product form solution with load independent exponential servers and where functions g i are defined by formula (11), i.e., for those closed network topologies and conditions that leads to formulas F1, F2, F4, F6 and F7. The convolution algorithm is illustrated in Figure 3. The algorithm is based on a set of recursive equations used to compute the normalizing constant G in formula (10), from which the marginal queue length distribution of each service center and the average performance indices are derived by closed-form expressions. For example the average queue length of node j, if AM-BM-aj>B(M)-Bj+aj, where Aj and aj are defined in Fig.3, can be written as follows: L j = 1 GM (N) B j k-aj  kr j [GM(N - k + a j) - rjgm(n - k + a j - 1)] k=a j and other performance indices such as node utilization, throughput and average response time are derived as defined in Section 2, by using product form expression (10) for the state probability. For a detailed description of the algorithm and the performance indices expression see [BDO01, BC98]. It has a linear time computational complexity in the number of network components: it requires O(M N) operations, and more precisely O(MC), where C = max {Bi-ai 1 i M} and ai is the minimum node i population defined in Fig.3. MVA An MVA algorithm has been defined for the particular case of product form networks with cyclic topology and with BBS-SO and RS blocking. In this case product form F2 solution holds when the condition 1 (non empty condition) is satisfied. The solution is based on the definition of a dual T6/14

15 [Computation of the values aj, Aj, B(j)] for j=1 to M { a j = max(0, N - Â Bk ) ; A j = Â ai ; B(j) = Â Bi } 1 k M, k j 1 i j 1 i j [initialisation] for n =a1 to B1 {G1(n) = r 1 n }; [computation of the functions Gj(n)] for j=2 to M {MIN = min (Aj-1+Bj, B(j-1)+aj); MAX = max (Aj-1+Bj, B(j-1)+aj;); G j(a j) = r j a j G j-1(a j-1); for n= Aj +1 to min(min, N) {G j(n) = r j aj Gj-1(n - a j) + rj G j(n - 1) }; for n= MIN +1 to min(max,n) if MIN = B(j-1)+aj then G j( n) = rj G j( n-1) else G j(n) = r j a j G j-1(n - a j) - r j B j+1 G j-1(n - Bj - 1) + rjgj(n - 1) ; for n=max +1 to min(b(j)-1, N) B {G j( n) j+1 = - r j G j-1( n-b j-1) + rj G j( n-1)}; if N > B(j) then G j(b(j)) = r j B j G j-1(b( j-1)) } Figure 3. Convolution algorithm for closed product form QNB. network without blocking with identical product form state distribution, as described in [GN67]. Then by using the duality property this MVA algorithm simply applies the standard MVA algorithm for networks without blocking to the dual network. A description of this MVA algorithm can be found in [Cl98]. Note that this MVA algorithm is not related to the arrival theorem as the MVA for queueing networks without blocking [RL80], since it is based on the dual network that is without blocking. The arrival theorem for networks with blocking is discussed in Section 4.3 and for product form networks with blocking in Section 5.4. The arrival theorem for product-form queueing networks with infinite capacity queues has been extended to some cases of QNB, but with a different interpretation. A discussion of the arrival theorem validity for product form QNB with different blocking types including Stop and Recirculate blocking and necessary and sufficient condition under this product form can be found in [BV97].The arrival theorem for special case of two-node product form exponential networks with BBS-SO or BAS blocking is proved in [BD89]. Another MVA algorithm has been extended to a class of product form networks and with RS- RD blocking, based on recursive relations for average performance indices on the network with blocking. It applies to networks with F2 or F3 product form solution and has a time computational complexity of O(B + M N) operations, where B + =max 1 i M Bi. A detailed description of this MVA algorithm can be found in [S99]. 3.4 Heterogeneous networks Heterogeneous QNB allow representing complex systems where different nodes can work under various blocking mechanisms. For example in an heterogeneous QNB modeling a computer network, RS-RD blocking can be used to model a system components that repeats the sending of customer (a message or a packet) when it fails due to finite buffer constraints, while BAS blocking T6/15

16 can represent a system component that is blocked after the service because of the full destination node. The exact solution of a heterogeneous queueing network with blocking (HQNB) can be obtained by the Markov chain approach and, in some special cases, by product form solution. The definition of the Markov process associated to the HQNB combines and composes the definition of the processes for the various blocking type. A complete definition of the state space of an HQNB can be found in [BDO01]. In this section for the sake of simplicity, we do not consider BBS-SNO, Stop and Recirculate mechanisms. Note that a node j working under a given blocking type X means that when its destination node i is full, node j behaves according to X blocking model. Then process definition is more complex because more constraints of the different blocking types must be verified for a given node as upstream nodes working under different blocking types. For example, consider that for any node i with finite capacity and at least an upstream node j with BAS blocking, the node state notation Si must include component mi as defined in Example 2. We can define the following partition of the set of M network nodes: U IX = { 1, 2, º, M} X Œ B where B = {Unb, RS-RD, BAS, BBS-SO, BBS-O}, IX is the set of the nodes with X blocking type and X=Unb denotes that a node is never blocked. The state space of a closed heterogeneous network with M nodes operating according to a blocking type in B is defined as follows: EHet = {(S 1, S 2,..., S M ) Si = (ni,mi) "iœ IRS-RD» IBBS-O» IUnb; S i = (ni,si,mi) M "iœ IBAS; S i = (n i,ns i,mi) "iœ IBBS-SO; ai ni Bi, 1 i M,  ni =N } i=1 where components si, mi and NS i are defined as in formulas (8) and (9) and we assume mi= if node i has not sending node with BAS blocking. Note that each job transition from node j to node i can yield simultaneous transitions if mj, and it has at least an upstream node with BAS blocking and a job waiting for room in node j. For an open network, the process state space definition is simplified by setting ai=0 in the state space definition. The process transition rate matrix Q is defined by an appropriate composition of the various condition given in the process definition of the different blocking type. A complete definition of the transition rate matrix of an HQNB can be found in [BDO01]. As concerns product form solution of HQNB, a few results have been obtained by using some equivalence properties between blocking types. Indeed, it can be proved that two networks with different blocking types can be equivalent in terms of queue length distribution and average performance indices under particular constraints, such as the type of service centers and the network topology [BD94, BDO01, P90, DLT94, OP86, OP88, OP89a]. Then one can proof that product form solution for HQNB hold in the cases illustrated in Figure 4 for various network topologies and blocking types, with the corresponding product form formula given by expressions (11) and (10). For example a cyclic network illustrated in Fig. 4 has product form solution F2 when each node can have RS-RD, RS-FD or BBS-SO blocking. T6/16

17 Product form formula F1 F2 & Cond1 F3 F5 F2 & Cond2 & Cond4 for BBS-SO and RS-FD Blocking types BAS BBS-SO RS-RD RS-FD BBS-SO RS-RD RS-FD BBS-SO RS-RD RS-FD node 1 with RS RS-RD Stop --- Network topology Two nodes Cyclic Central server Reversible routing Arbitrary Figure 4. Product form for heterogeneous networks: formulas and admitted blocking types for each network topology. 4. Approximate analysis of closed queueing networks with blocking When exact analysis of QNB cannot be applied because of the state space explosion of the Markov chain and because the product form conditions do not hold, we can consider an approximate analysis technique. Many approximate methods to analyze open or closed QNB have been proposed to evaluate average performance indices and queue length distributions [BDO01, O93, P89]. Most of the method provides an approximate solution with a limited computational cost, but any bound on the introduced approximation error. The accuracy of the methods is usually validated by comparing numerical results with either simulation results or exact solutions. 4.1 Basic principles Several approximate methods are heuristics based on the decomposition principle applied to the underlying Markov process or directly to the network. Decomposing a Markov process consists in identifying a state space E partition of into K subsets Ek, 1 k K, which leads to a decomposition of the rate matrix Q into K2 submatrices. Hence the solution of the system (6) is reduced to the solution of K subsystems of smaller dimension, each related to a subset of E. Then these solutions are combined to obtain the overall process solution. The state space E partition is a critical problem and affects the accuracy and the time complexity of the approximate algorithm. The decomposition principle applied to the QNB is based on the aggregation theorem for queueing networks. It performs in three steps: (I) network decomposition into a set of subnetworks, (II) analysis of each subnetwork in isolation to define an aggregate component, (III) definition and analysis the new aggregated network. Then the analysis of the original network reduces to the analysis of each subnetwork at step II and of the aggregated one at step III. In order to define efficient methods, the isolated subnetworks at step II and the aggregated network have to be simple to analyze. The aggregation T6/17

18 theorem provides exact results for product-form QNB, i.e., the aggregated network is equivalent to the original model in terms of queue length distribution and average performance indices. For non product-form networks the aggregated network only approximates the original model, and no bound is known on the approximation error. Various heuristics have been defined by taking into account both the network model characteristics and the blocking type [AP89, AP87, BK81, BJ88, DF89, DF93, FD89, G87, GN67, HB67, KR78, J90, KX89, KD93, KA95, KA03, LBDF95, O90, O93, PA86, P89, PNL88, PS89, SD86, YB85, B00]. A bounded aggregation technique has been defined for Markov processes and applied to queueing networks with blocking in [CS86] by exploiting the special structure of the underlying Markov process. Network decomposition and the evaluation of the approximation error are the two main critical issues for the approximate methods based on decomposition. Some techniques apply an iterative aggregation-disaggregation procedure for which conditions and speed of convergence should also be considered, as in [DF89]. Some approximations are obtained by forcing exact aggregation for product-form queueing networks with infinite capacity. Other approximation algorithms are based on a product-form solution defined by the maximum entropy principle [KX89, KD93, KA95]. We briefly present some approximate methods for closed QNB in this Section and for open QNB in the next section. For simplicity we assume single server nodes and FCFS service discipline. Figure 5 shows some approximate methods for closed QNB for cyclic and arbitrary topology networks and for some blocking types. Blocking type BAS - Throughput Approximation - Network Decomposition - Matching State Space - Approximate MVA RS-RD BBS-SO - Throughput Approximation - Variable Queue Capacity Decomposition - Maximum Entropy Algorithm Cyclic Arbitrary Network topology Figure 5. Approximate methods for closed QNB. Three methods analyze network with cyclic topology and the three approximations apply to arbitrary topology networks. All of them assume exponential service time distribution, except for the Maximum Entropy Approximation that applies to QNB where nodes have generalized exponential service time distribution (GE). All the algorithms apply to homogeneous networks, i.e., networks where each node has the same blocking type. Let B=S 1 i M Bi denote the total network capacity, and B + and B - respectively the maximum and minimum node capacity of the network. T6/18