Queueing Networks with Blocking analysis, algorithms and properties

1. Queueing Networks with Blocking • analysis, algorithms and properties Simonetta Balsamo Università Ca’ Foscari di Venezia Dipartimento di Informatica Venice, Italy

2. Outline • Queueing networks with blocking • • Models of systems with • finite capacity resources - population constraints • • Types of blocking mechanisms • various system behavior (network protocols,technologies) • • Performance indices • average (throughput, utilization, mean response time) • distribution (queue length, blocking probability, effective throughput) • • Analytical solution methods •  exact solution •  approximate solution methods •  solution algorithms, comparison, conditions • • Some equivalence properties • • Some application examples • • Open research I) II) III) IV)

3. n≤ B n≤B Queueing networks with blocking (QNB):finite capacity queues (I) • Queueing networks represent • resource sharingand contention by a set of customers • Queueing networks with blocking consider • resources with finite capacity queues • population constraints • finite capacity of the queue • n number of customers in the service center • B finite capacity • blocking  dependence • deadlock • various blocking types: • different behaviors of customer arrivals at a full node and of servers' activity • heterogeneous QNB: • service centers may have different blocking types

4. n≤B n≤B … … Queueing networks with blocking (QNB):finite population constraint • • (sub)network population constraint • n number of customers in the network • B network finite capacity • if n=B then arrivals are lost • blocking  dependence • deadlock • QNB analysis: exact • approximate methods • simulation

5. Blocking Types • various blocking types: • different behaviors of customer arrivals at a full node and of servers' activity • Queueing networks with finite capacity queues: • BASBlocking After Service • BBSBlocking Before Service • RSRepetitive Service Blocking • Queueing networks with (sub)network population constraint: • STOP • Recirculate

6. nj≤ B i j Blocking Types • QNB with finite capacity queues • BASBlocking After Service • BBSBlocking Before Service • RSRepetitive Service Blocking • Blocking • After • Service • if a job after its service attempts to enter a full node, is forced to wait in front of the sending server; the service is blocked until the job enters the destination node • unblocking scheduling • First Blocked First Unblocked

7. nj≤ B i j nj≤ B j i Blocking Types • Blocking • Before • Service • a job declares its destination node before its service; if the destination is full, the server is blocked until a departure occurs from the destination node. • If the destination node becomes full, the service is interrupted and the server is blocked; the destination does not change • BBS-SO vs BBS-SNO (Server Occupied or Not) • Repetitive • Service • Blocking • if a job after its service attempts to enter a full node, is forced to repeatthe service in the sending node • RS-RD vs RS-FD (Random or Fixed Destination)

8. L≤n≤U n≤B … … Blocking Types • QNB with population costraints • n[L,U] • a(n) = 0 for n≥U load dependend arrival rate • d(n) = 0 for n≤L load dependend service rate • STOP blocking • if d(n) = 0 then service at each node is stopped • Service is resumed upon a new arrival to the network. • RECIRCULATE Blocking • a job upon completion of its service at node i, leaves the network with probability pi0 d(n), and it is forced to stay in the network with probability pi0 [1-d(n)], where pi0 is the routing probability. • That is, a job adter its service at node i enters node j with state dependent routing probability pij + pi0 [1-d(n)] p0j, 1≤i,j≤M, n≥0.

9. Deadlock • In queuing networks with finite capacity deadlock can occur with • BAS , BBS , RS-FD • Prevention or detection and resolving techniques • A simple prevention technique • and for BAS and BBS pii = 0 for each node i • NOTE • networks with finite capacity and RS-RD with irreducible routing matrix and the network population is less than the total buffer capacity do not deadlock Total buffer capacity of the queues in each possible cycle of the network the overall network < population

10. QNB: performance indices • Related to a single resource i (a service center) • average indices • random variables • Ninumber of customers in the resource • ticustomer passage time through the resource • distribution of ni • πi(ni)at arbitrary times •  (ni)at arrival times of a customer at the resource • Related to the overall network • average indices • passage time • job loss probability (for open networks)

11. Notation - QNB • Network model parameters • M number of nodesl total arrival rate • Nnumber of customers (closed network) µiservice rate of node i • P=||pij||routing matrixp0iarrival probability at node i • xivisit ratio at node i, solution oftraffic equations • Bi finite capacity of node i bi(ni) blocking function • 0<bi(ni)≤1, for 0≤ni<Bi, bi(Bi)=0 • Performance indices • depend on the blocking type • are derivedfromthestate probabilityπi(ni)or (ni) xi = l p0i + j xj pji

12. Performance indices • For single server node i • utilization Ui = 1 - πi(0) - PBi • throughput Xi = ni[ πi(ni) - PBi(ni)] µi(ni) • Xi = Ui µifor constant service rate • mean queue length Li = nini πi(ni) • mean response time Ti = Li /Xi • mean cycle time for node i j xj Tj /xi • PBi(ni) probability that node i is not empty and blocked when there are ni customers in i • PBi =niPBi(ni) overall blocking probability • PB definition depends on theblocking type • Effective utilization when the server are neither empty nor blocked • Effective throughput the useful work (for RS and BBS)

13. Model Analysis Approximate analysis Exact analysis Open QNB Closed QNB Markov chain generation Product-form algorithms Approximate algorithm selection Approximate algorithm selection Markov chain solution Model constraints (on topology, blocking type,…) Model constraints (on topology, blocking type,…) Model constraints (on topology, blocking type,…) Computation of performance indices Computation of performance indices Analytical solutions for QNB • Evaluation of average performance indices and joint queue length distribution at arbitrary times (π) • • exact solution • based of Markov process analysis • product-form solution of π • • approximate and bound solution

14. Analytical solutions for QNB (II) • S = (S1,…,SM) system state • Si node i state which includes ni , 1≤i≤M • E set of all feasible states • E discrete state space • Q infinitesimal generator • if P (network routing matrix) irreducible • then  ! stationary state distribution π = {π(S), SE} • solution of the global balance equations • • the definition of S, E and Q depends on •  the network characteristics •  the blocking type of each node Markovian network the network behavior can be represented by a homogeneous continuous time Markov process M π Q = 0 , SEπ(S) = 1

15. n2 ≤ B2 n1 ≤ B1 m m 1 2 A simple example: two-node cyclic network • FCFS service discipline • exponential service time • S = (S1,S2) system state definition • Si = niRS or BBS blocking • if ni = BiRS server active, BBS server blocked • Si = (ni, si) BAS blocking • where si is the server state: si=1 (active) si=0 (blocked) • birth-death Markov process • closed-form solution

16. A simple example: two-node cyclic network • Let  = (µ1/µ2) • for BBSandRS • π(S)=(1/C)n2-N+B1 S= (n1, n2)E C=S0≤i≤B1+B2-Ni • for BAS • π(S)=(1/C)n2-N+B1+1S=((n1,1),(n2,1)) E • π(S)=(1/C) S=((B1,1),(N-B1,0)) • π(S)=(1/C)B2+B1+2-N S=((N-B2,0),(B2,1)) • C=S0≤i≤B1+B2+2-Ni • for infinite capacity queues (no blocking) • π(n1, n2)=(1/C)n20≤n1≤N , n2=N-n1 C=S0≤i≤B1+B2+2-Ni

17. Analytical solutions for QNB: state definition • S=(S1,…,SM) system state • Sistate of node i which includesni, 1≤i≤M • exponential network, First Come First Served discipline, general topology • dithe destination node of the next job that will exit from node i • siserver state: active (1) blocked (0) • mi= (mi,…,mu(i)), 0≤u(i)≤M-1 • queue of indices of the nodes blocked by node i, if ni=Bi • unblocking scheduling • RSthe server is always active • BBS-SOthe server is blocked if ni>0 and ndi=Bdi • BBS-SNOidem and ni<Bi • BBS-O the server is blocked if at least one of the destination nodes of node i is full ( j : pij>0 and nj=Bj)

18. Analytical solutions for QNB: process definition • • different process transition rate matrices Q dependent on blocking type • Q= ||q(S,S')|| • RS-RD • q(S,S') = (nj) µj bi(ni) pji if S'= S + ei - ej • q(S,S') =  (nj) µj pj0 if S'= S - ej • q(S,S') =  p0j bj(nj)if S'= S + ej •  total arrival rate • bi(ni) blocking function of node i • (ni)=0 if ni=0, (ni)=1 otherwise, 1≤i≤M • ei M-vector with all zero except one in i-th position • q(S,S) = - S'E, S'≠Sq(S,S')

19. Exact analysis of Markovian QNB • Solution algorithm for the evaluation of average performance indices and joint queue length distribution at arbitrary times (π) in Markovian QNB 1 Definition of system state and state space E 2 Definition of transition rate matrix Q 3 Solution of global balance equations to derive π 4 Computation from π of the average performance indices • This method becomes unfeasible as |E| grows, • i.e., proportionally to the dimension of the model (number of customers, nodes and chains) • exact product-form solution under special constraints • approximatesolution methods

20. Exact analysis of QNB: special cases • subset of Markovian networks • product-form solution of π • single class open or closed networks under certain constraints, depending on the network definition and the blocking type • G normalizing constant • n total network population • V and gi depend on • network parameters (x, µi) and population • blocking type • additional constraints • Various formulae F1-F5 define functions V and gi for different combinations of • network topology • blocking type • Computationally efficient exact solution algorithms • Convolution Algorithm • Mean Value Analysis

21. Central server (star) Reversible routing Two nodes Cyclic Arbitrary BAS BBS-SO RS-RD RS-FD BBS-SO RS-RD RS-FD node 1 with RS BBS-SO RS-RD RS-FD RS-RD Stop BAS BBS-SO RS-RD RS-FD F2 & Cond 2, 4 for BBS-SO and RS-FD F7 & Cond 5 for BAS F2 & Cond1 F1 F3 F5 Product-form heterogeneous QNB • Formulas and admitted blocking types for each network topology, with additional constraints Network topology Blocking types Product form formula Product-form conditions Product-form formulas

22. QNB Product-form: constraints and formulas A-type node: arbitrary service time distribution, symmetric scheduling discipline or exp. service time, identical for each class at the same node, when the scheduling is arbitrary.

23. QNB Product-form: constraints and formulas Formula F3 multiclass central server networks with the class type of a job fixed in the system state-dependent routing depending on the class type blocking functions dependent on node and class A-type nodes For single class exponential networks with load dependent service rates µi(ni)=µifi(ni) state-dependent routing p1j(nj)= wj(nj) w(N-n1) nj, pj1=1 for 2≤j≤M, ni , 1≤i≤M

24. Exact analysis of QNB: product-form principles • most of the product-form solutions have been derived by applying • reversibility of the underlying Markov process • duality • reversibility • the underlying Markov process of the QNB can be obtained by truncating the reversible Markov process of the network with infinite capacity (by the theorem on truncated Markov process): the same solution as the whole process normalized on the truncated sub-space holds  product-form solution • examples • - two-node exponential single class cyclic networks • - multiclass networks with BCMP, RS blocking and reversible routing P • P is reversible  xipij = xjpjii,j

25. Exact analysis of QNB: product-form principles • duality • a dual network is obtained from the original one by reversing the connections between the nodes • Not-Empty-Condition of original network dual network without blocking • (product-form) • examples • - exponential cyclic network with BBS or RS • - arbitrary topology networks with load independent service rates for RS-RD blocking • - closed cyclic networks with phase-type (general) service distributions and BBS-SO blocking for whichthe throughput of the network is shown to be symmetric with respect to its population • B=SiBi X(N-B) = X(B)

26. Product-form QNB: algorithms • Algorithms forclosedQNB • Polynomial time computational complexity • Convolution: evaluation of the normalizing constant and average performance indices • MVA: direct computation of average performance indices (mean response time, throughput, mean queue length) • Convolution • RS and BBS blocking, arbitrarytopology, load independent service rates • F1 or F2 product form solution • based on a set of recursive equations, derivation of • - marginal queue length distribution πi(ni) • - mean queue length Li • - mean response time Ti • - throughput Xi • - utilization Ui • - mean busy period • - blocking probabilities • computational complexity: O(M N) • Specifically: O(M C) • C=max{Bi - ai, 1≤i≤M} ai minimum feasible queue length of node i Algorithm

27. Product-form QNB: algorithms • MVA: direct computation of average performance indices • RS blocking, cyclic topology, load independent service rates • F2 product form solution • equivalence properties: • dual network without blocking • based on the MVA algorithm for the dual network • derivation of • - mean queue length Li • - mean response time Ti • - throughput Xi • - utilization Ui • - mean busy period • - blocking probabilities • computational complexity: O(M N)

28. Exact analysis of QNB: special cases • symmetrical networks • identical blocking type, • identical values of µi and Bi for each node i • routing P where all rows are identical up to a rotation of the entries • exponential networks • efficient computation of π and average indices reduction algorithm based on exact aggregation of the Markov process, due to the special network structure • - identification of a partition of E in K subsets {Ek , 1≤k≤K} • - decomposition-aggregation procedure • π(S) = Prob(S | Ek) πa(Ek) • - for symmetrical networks: uniform conditional distribution • Prob(S | Ek) = 1/ #Ek • - aggregated probabilitiies πa= πaQa • - with aggregated matrixQa= || qa(k,h) || • qa(k,h) = (1/ #Ek) Qkh 1T where 1= (1,…,1) • computation of πreduces to the computation ofπa:O(K3 )

29. Example of symmetrical networks • Each node has the same probabilistic behavior • µi = µ, Bi = B, 1≤i≤M • p1i ≠0  p1+m ((I+m-1)mod M)+1 ≠0 , 1≤i≤M ,1≤m≤M-1 • pi j ≠0 andpi k ≠0  pi j =pi k = r, 1≤i,j,k≤M • where r=1/K, if K is the outdegree of each node • exponential service time, abstract service discipline (FCFS) • Examples of symmetrical network topologies

30. Approximate analysis of QNB (III) • Many approximation methods • Most of them do not provide any bound on the introduced error • Validation by comparison with exact solution or simulation • Basic principles • - decompositionapplied to the Markov process or to the network • - forced product-formsolution • - structural properties for special cases • - maximum entropy • Various accuracy and time computational complexity

31. Markov Process Decomposition • Markov process with state space E and transition matrix Q • • Identify a partitionof Einto K subsets • E=U1≤k≤KEk •  decomposition of Q • • decomposition-aggregation procedure • Prob(S|Ek)conditional distribution • πaaggregated probabilities • • computation of π(S) reduces to • the computation of Prob(S | Ek)  S, Ek • the computation of πa • • exact computation soon becomes computationally intractable • EXCEPT FOR special cases (symmetrical networks) • • approximationof Prob(S | Ek) and Prob(Ek) π(S)= Prob(S | Ek)πa(Ek)

32. Process and Network Decomposition • Heuristics take into account • the network model characteristics • the blocking type • NOTE: the identification of an appropriate state space partition affects • the algorithm accuracy • the time computational complexity • If the partition of E corresponds to a NETWORK partition into subnetworks  network decomposition subsystems are (possibly modified) subnetworks • The decomposition principle applied to QNB • is based on the aggregation theorem for QNB • 1. network decomposition into a set of subnetworks • 2. analysis of each subnetwork in isolation to define an aggregate component • 3. definition and analysis of the new aggregated network

33. Network Decomposition • 1. network decomposition • NP-complete problem: critical issue • 2. analysis of isolated subnetworks • choose simple subnetworks • apply efficient solution methods • 3. aggregated network analysis • aggregation theorem: • exact only for product-form networks • approximation otherwise • unknown error Various approaches determine the subnetwork parameters

34. Network Decomposition • • approximations based on the forced application of the exact aggregation technique for product-form QN without blocking • low computational cost • accuracy: experimental results • suitable for many practical cases • BUT the approximation error is UNKNOWN • • many approximations are based on iterative solution of subsystems or subnetworks • Iterative aggregation-disaggregation • speed and proof of convergence • • few approximate techniques with known accuracy • bound solutions can be used as approximation methods with known accuracy • • open issue: solution of general classes of heterogeneous QNB

35. Approximate methods for QNB • Method comparison • - model assumptions • - algorithm rationale • constraints on the network parameters • topology, service distribution,blocking type • - performance comparison • accuracy • efficiency • class of models to which they can be applied • - model parameters • nodes, customers, topology, service rates, queue capacity • - symmetry of network parameters • Six significant approximate methods for closed QNB • Four significant approximate methods for open QNB Experiments

36. Approximate methods forclosedQNB • M exponential, G general, GE generalized exponential • A/B/s Kendall’s notation: • A customer interarrival time distribution • B service time distribution • s the number of identical servers

37. n1 ≤ B1 n2 ≤ B2 nM ≤ BM … 2 M 1 m m m 1 2 M N jobs Cyclic Networks • Exponential service times • Performance index: • network throughput • as a function of the network population: X(N) Methods

38. Arbitrary closed topology QNB • MSS and AMVA assume networks with exponential service time and evaluate the network throughput • ME Algorithm assumesgeneralized exponential service timeandevaluates the queue length distribution and average performance indices Methods

39. Algorithm forclosedQNB: comparison Observations

40. Approximate methods foropenQNB • The approximation principle is network decomposition for all the algorithms • One-node subneworks as • M/Cox/1/B queue by Tandem Phase-Type Decomposition • M/M/1/B queue by the other algorithms • Last algorithm applies the maximum entropy principle Methods

41. n1 ≤ B1 n2 ≤ B2 nM ≤ BM … 2 1 M i T(i) Tandem Networks • BAS blocking • Exponential service times • Performance index: network throughput • Tandem Exponential Decomposition and • Tandem Phase-Type Decomposition apply network decomposition • M one-node subnetwork T(i), 1≤i≤M • T(i) corresponds to node i • analysis of isolated subnetworks • T(i) as aM/M/1/Biqueue TED • T(i) as aM/PHn/1/Biqueue TPD • efficient solution methods Methods

42. Algorithm foropenQNB: comparison • All algorithms evaluate for each node i • πi queue length distribution • Li mean queue length, Xi node throughput, Ri node mean response time Observations

43. QNB: equivalence Properties • equivalenciesin terms of • - stateprobability distributionπ • - average performance indices • passage time distribution • most of the equivalencies derive from the identity of the network processesÞπ • Remarkeven if two networks have identical Markov processes, the meaning of corresponding states may be different • Þ performance measures may be NOT equivalent • Þ equivalence in terms of π does NOT necessarily lead to equivalence in terms of average performance indices • extension of efficient computational algorithms (MVA and Convolution)andsolution methods to QNB (e.g.: aggregation technique) • equivalence between networks •  with and without blocking •  with different blocking types •  homogeneous and non-homogeneous networks

44. Equivalence between networks with and without blocking (IV) • Parameters of the network with infinite capacities • µ*if*i(ni) load dependent service rates, P* routing matrix • π* state distribution • exponential networks with RS-RD blocking • f*i(k) any positive arbitrary function for k>Bi • hi = ei yi , ei defined in formula F2 • y=(y1,…,yM), y=yA, A=||aij||, aij=pji , j≠i, aii=1-Sj≠i aij 1≤i,j≤M

45. Equivalence between networks with different blocking types X and Y blocking types X=Y identity identical π X®Y reducibility correspondence between π usually with modified capacities BiX finite capacity when node i works under blocking type X (I) multiclass networks, BCMP type nodes, class independent capacities (II) single class networks, exponential nodes, load independent service rates

46. Equivalence between networks with different blocking types: closed QNB

47. Equivalence between networks with different blocking types: open QNB

48. Equivalence between heterogeneous QNB REMARK non-homogeneous networks where nodes work under different and equivalent blocking types are also equivalent to homogeneous networks with one of the blocking types • extension of solution methods to QNB - exact analysis - approximate algorithms

49. Application example of QNB • Store-and-forward packet switching networks • Circuit switching networks • - data packets •  travel through the network or wait to be transmitted •  routing • - system resources •  shared by the data to be transmitted • - network topology • - allocation of link capacity for the connection (circuit switching) • Problems • - Buffer allocation • Determine the amount of buffer space to be allocated to each station to optimize system performance (e.g. maximize network throughput, minimize end-to-end delay) • - Routing algorithm • - Scheduling • Performance measures • - Average packed delay over the entire network • - End-to-end delay for pairs source-destination • - Buffer occupancy • - Loss probability

50. Application example of QNB: communication networks • A model of a store-and-forward packet switching network with virtual circuits • level 3 in OSI reference model • Independence assumptions • Window flow control • Closed cyclic network, RS blocking • Stations and network nodes have finite buffer • Performance indices network throughput, delay, buffer occupancy N packets window size