Queueing Networks with Blocking analysis, algorithms and properties

1PERFORM-QNMs ‘06 - HET-NETs ‘06S. Balsamo

Università Ca’ Foscari di Venezia

Queueing Networks with Blocking

analysis, algorithms and properties

Simonetta BalsamoUniversità Ca’ Foscari di Venezia

Dipartimento di InformaticaVenice, Italy



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

Outline

I)

II)III)

IV)



Queueing networks represent resource sharing and 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 dependencedeadlock

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

Queueing networks with blocking (QNB):finite capacity queues

n≤ Bn≤B

(I)



• (sub)network population constraint

n number of customers in the network B network finite capacity

if n=B then arrivals are lost

blocking dependencedeadlock

QNB analysis: exactapproximate methodssimulation

Queueing networks with blocking (QNB):finite population constraint

n≤B

……

n≤B



various blocking types:

different behaviors of customer arrivals at a full node and of servers' activity

Queueing networks with finite capacity queues:

BAS Blocking After Service BBS Blocking Before ServiceRS Repetitive Service Blocking

Queueing networks with (sub)network population constraint:

STOPRecirculate

Blocking Types



QNB with finite capacity queues

BAS Blocking After Service BBS Blocking Before ServiceRS Repetitive 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 schedulingFirst Blocked First Unblocked

nj≤ B

ji

Blocking Types

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.



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 changeBBS-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 repeat the service in the sending nodeRS-RD vs RS-FD (Random or Fixed Destination)

ji

nj≤ B


ji

nj≤ B


Blocking Types



QNB with population costraintsn[L,U]

a(n) = 0 for n≥U load dependend arrival rated(n) = 0 for n≤L load dependend service rate

STOP blocking

if d(n) = 0 then service at each node is stoppedService 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.

Blocking Typesn≤B

……

L≤n≤U



In queuing networks with finite capacity deadlock can occur withBAS , BBS , RS-FD

Prevention or detection and resolving techniquesA 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

the overall networkpopulation <

Total buffer capacityof the queues

in each possiblecycle of the network

Deadlock



Related to a single resource i (a service center) average indices

random variables Ni number of customers in the resourceti customer 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 networkaverage indices

passage timejob loss probability (for open networks)

Ui utilizationXi throughputLi mean queue lengthTi mean response time

U utilizationX throughputL mean population

(for open networks)T mean response time

QNB: performance indices



Network model parametersM number of nodes total arrival

rateN number of customers (closed network) µi service rate of

node iP=||pij|| routing matrix p0i arrival probability

at node ixi visit ratio at node i, solution of traffic 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 typeare derived from the state probability πi(ni) or (ni)

xi = p0i + j xj pji

Notation - QNB



For single server node i utilization Ui = 1 - πi(0) - PBi

throughput Xi = ni [ πi(ni) - PBi(ni)] µi(ni)

Xi = Ui µi for constant service rate

mean queue length Li = ni ni 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 n i customers in i

PBi =ni PBi(ni) overall blocking probability

PB definition depends on the blocking type

Effective utilization when the server are neither empty nor blockedEffective throughput the useful work (for RS and BBS)

Performance indices



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

Analytical solutions for QNB

Computation of performance indices

Product-form algorithms

Markov chain solution

Markov chain generation

Model constraints(on topology, blocking type,…)

Model Analysis

Exact analysis Approximate analysis

Approximate algorithmselection

Approximate algorithm selection

Computation of performance indices



Open QNB Closed QNB



S = (S1,…,SM) system state Si node i state which includes ni , 1≤i≤ME set of all feasible states

E discrete state spaceQ infinitesimal generator

if P (network routing matrix) irreducible then ! stationary state distribution π = {π(S), SE}

solution of the global balance equations

• the definition of S, E and Q depends on the network characteristics the blocking type of each node

Analytical solutions for QNB

Markovian network the network behavior can be represented by a

homogeneous continuous time Markov process M

π Q = 0 , SE π(S) = 1

(II)



FCFS service disciplineexponential service timeS = (S1,S2) system state definition

Si = ni RS or BBS blocking if ni = Bi RS server active, BBS server blocked

Si = (ni, si) BAS blockingwhere si is the server state: si=1 (active) si=0 (blocked)

birth-death Markov processclosed-form solution

B

2

B

2

N-,, B

2

-1B

2

N- +1B

1

B

1

N-, B

1

B

1

N-,

-1 +1

μ

1

μ

1

μ

1

μ

1

μ

2

μ

2

μ

2

μ

2

…

μ

1

μ

2

μμ 1 2

n1 ≤ B1 n2 ≤ B2

A simple example: two-node cyclic network



Let = (µ1/µ2)

for BBS and RS

π(S)=(1/C) n2-N+B1 S= (n1, n2 ) E C=0≤i≤B1+B2-N i

for BAS

π(S)=(1/C) n2-N+B1+1 S=((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= 0≤i≤B1+B2+2-N i

for infinite capacity queues (no blocking)

π(n1, n2)=(1/C) n2 0≤n1≤N , n2=N-n1 C= 0≤i≤B1+B2+2-N i

A simple example: two-node cyclic network



S=(S1,…,SM) system stateSi state of node i which includes ni, 1≤i≤Mexponential network, First Come First Served discipline, general topology

di the destination node of the next job that will exit from node isi server 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

RS the server is always active BBS-SO the server is blocked if ni>0 and ndi=Bdi

BBS-SNO idem 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)

RS-RD Si = niBBS-O

BBS-SOBBS-SNO Si = (ni, di)

RS-FDBAS Si = (ni, si, mi)

Analytical solutions for QNB: state definition



• different process transition rate matrices Q dependent on blocking typeQ= ||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'≠S q(S,S')

Analytical solutions for QNB: process definition



1 Definition of system state and state space E2 Definition of transition rate matrix Q3 Solution of global balance equations to derive π 4 Computation from π of the average performance indices

Solution algorithm for the evaluation of average performance indices and joint queue length distribution at arbitrary times (π) in Markovian QNB

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 approximate solution methods

Exact analysis of Markovian QNB



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 populationblocking typeadditional constraints

Various formulae F1-F5 define functions V and gi for different combinations of network topology blocking type

Computationally efficient exact solution algorithmsConvolution AlgorithmMean Value Analysis

€

π(S) =1G V(n) gii=1

M∏ (n i )

Exact analysis of QNB: special cases



Formulas and admitted blocking types for each network topology, with additional constraints

Network topology

Blocking types

Product form

formula

Central server(star)

Reversible routingTwo

nodes CyclicArbitrary

BASBBS-SORS-RDRS-FD

BBS-SORS-RDRS-FD

node 1 with RS

BBS-SORS-RDRS-FD

RS-RDStop

BASBBS-SORS-RDRS-FD

F2& Cond 2, 4for BBS-SO and RS-FD

F7& Cond 5

for BAS

F2& Cond1F1 F3 F5

Product-form heterogeneous QNB

Product-form conditions Product-form formulas



QNB Product-form: constraints and formulasFormula Conditions V(n) gi(ni), ∀ni,∀i

F1μ uticassnetwoksBCMPtyπenoescass ine πene ntcaπacities

F5 asF4,butsingecass 1 (i/µi)ni

F7μ uticassnetwoksFCFe πonen tiano escassine π.caπacities

F4

μ uticassnetwokswithcasstyπefiebockingfunctionse π.onnoe ,cassanchainAtyπenoesoae π.seviceatesµi(ni)=µifi(ni)

1

(i/µi)ni.

. b

i

( l − 1 )

f

i

( l )l = 1

n

i

∏

F2

s ingle class Q N, exp. nod e s ,loa d inde p. serv ice rat e swith ε =(ε 1,…, ε M)ε = ε P', P' = | | p 'ij | |p'ij=µjpji, i≠j,p'ii=1-Σj≠ip'ji, 1≤i,j≤M

1 1 / εini

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.



QNB Product-form: constraints and formulasFormula F3 multiclass central server networks with the class type of a job fixed in the systemstate-dependent routing depending on the class typeblocking functions dependent on node and classA-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€

V(N) = w(l−1)

l=1

N −n1

∏ wj ( l−1)

l=1

nj

∏j=2

M

∏

€

gi (ni ) =1

μ i

bi ( l−1)

f i (l )l=1

ni∏



most of the product-form solutions have been derived by applyingreversibility of the underlying Markov processduality

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 xi pij = xj pji i,j

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 which the throughput of the network is shown to be symmetric with respect to its population

B=iBi X(N-B) = X(B)

Exact analysis of QNB: product-form principles



Algorithms for closed QNBPolynomial 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)

ConvolutionRS and BBS blocking, arbitrary topology, load independent service ratesF1 or F2 product form solutionbased 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

Product-form QNB: algorithms

Algorithm



MVA: direct computation of average performance indices

RS blocking, cyclic topology, load independent service ratesF2 product form solutionequivalence 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)

Product-form QNB: algorithms



symmetrical networksidentical 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 entriesexponential 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 = πa Qa

- with aggregated matrix Qa = || qa (k,h) ||

qa (k,h) = (1/ #Ek) Qkh 1T where 1= (1,…,1) computation of π reduces to the computation of πa : O(K3 )

Exact analysis of QNB: special cases



Each node has the same probabilistic behaviorµi = 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 ≠0pi j =pi k = r, 1≤i,j,k≤Mwhere r=1/K, if K is the outdegree of each nodeexponential service time, abstract service discipline (FCFS)

Examples of symmetrical network topologies

Example of symmetrical networks



Many approximation methods Most of them do not provide any bound on the introduced errorValidation by comparison with exact solution or simulation

Basic principles- decomposition applied to the Markov process or to the network

- forced product-form solution - structural properties for special cases- maximum entropy

Various accuracy and time computational complexity

Approximate analysis of QNB(III)



Markov process with state space E and transition matrix Q • Identify a partition of E into K subsets

E=U 1≤k≤K Ek

decomposition of Q• decomposition-aggregation procedure

Prob(S|Ek) conditional distribution

πa aggregated probabilities

• computation of π(S) reduces tothe computation of Prob(S | Ek) S, Ek

the computation of πa

• exact computation soon becomes computationally intractable EXCEPT FOR special cases (symmetrical networks)

• approximation of Prob(S | Ek) and Prob(Ek)

π(S) = Prob(S | Ek) πa (Ek)

Markov Process Decomposition



Heuristics take into accountthe network model characteristicsthe blocking type

NOTE: the identification of an appropriate state space partition affectsthe 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 subnetworks2. analysis of each subnetwork in isolation to define an aggregate component3. definition and analysis of the new aggregated network

Process and Network Decomposition



1. network decomposition NP-complete problem: critical issue

2. analysis of isolated subnetworkschoose simple subnetworksapply efficient solution methods

3. aggregated network analysisaggregation theorem:exact only for product-form networks

approximation otherwiseunknown error

Various approaches determine the subnetwork parameters

Network Decomposition



• approximations based on the forced application of the exact aggregation technique for product-form QN without blocking

low computational cost accuracy: experimental resultssuitable for many practical cases

BUT the approximation error is UNKNOWN

• many approximations are based on iterative solution of subsystems or subnetworks Iterative aggregation-disaggregationspeed and proof of convergence

• few approximate techniques with known accuracybound solutions can be used as approximation methods with known accuracy

• open issue: solution of general classes of heterogeneous QNB

Network Decomposition



Method comparison - model assumptions- algorithm rationale

constraints on the network parameterstopology, service distribution,blocking type

- performance comparison accuracyefficiency 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

Approximate methods for QNB

Experiments



M exponential, G general, GE generalized exponential A/B/s Kendall’s notation:

A customer interarrival time distributionB service time distributions the number of identical servers

Method Network Constraints BlockingTopology Node type Type

ThroughputApproximation

cyclic G/M/1/B BAS orBBS-SO

NetworkDecomposition

cyclic G/M/1/B BBS-SO

Variable QueueCapacity

Decomposition

cyclica node withunlimitedcapacity

G/M/1/B BBS-SO

Matching StateSpace

general G/M/1/B BAS

ApproximateMVA

general G/M/1/B BAS

MaximumEntropy

Algorithm

general G/GE/1/B RS-RD

Approximate methods for closed QNB



Exponential service timesPerformance index: network throughput as a function of the network population: X(N)

Method BlockingType

Key idea

ThroughputApprox.[Onvural-Perros’89]

BAS orBBS-SO

Exact model analysis forsome network population.Throughput interpolation byvarying network population.

NetworkDecomp.

[Frein-Dallery’89]

BBS-SO Network decomposition intosingle nodes analyzed inisolation as M/M/1/B queues

VariableQueue

CapacityDecomp.

[Suri-Diehl’86]

BBS-SO Network aggregation of theset of finite capacity queuenodes in a single compositenode having state dependentservice rate and variablebuffer size

1 2 M

μ1 μ2

…

N jobsμM

n1 ≤ B1 n2 ≤ B2 nM ≤ BM

Methods

Cyclic Networks



MSS and AMVA assume networks with exponential service time and evaluate the network throughput

ME Algorithm assumes generalized exponential service time and evaluates the queue length distribution and average performance indices

Method BlockingType

Key idea

MatchingState Space

[Akyildiz’88]BAS

Analysis of the QNwithout blocking bychoosing the networkpopulation toapproximately match thestate space cardinality

ApproximateMVA

[Akyildiz’88]BAS

Modified MVA algorithm toconsider blocking

MaximumEntropy

Algorithm[Kouvatsos-Xenios’89]

RS-RDApproximate product-formfor the queue lengthdistribution

Methods

Arbitrary closed topology QNB



Method PerformanceIndices

Accuracy Efficiency

ThroughputApprox.

X(N) networkthroughputas a function ofthe population

Very good Poor fornetworkswith morethan 5 nodes

NetworkDecomp.

X networkthroughput

Good Good

VariableQueue

CapacityDecomp.

X(N) networkthroughputas a function ofthe population

Good fornetworkswith up to 4nodes,inaccurateotherwise

Fair

MatchingState Space

Xi nodethroughput

Fair Good

ApproximateMVA

Li mean queuelengthXi nodethroughputRi node meanresponse time

Fair forthroughput,poor forotherperformanceindices

Very good

MaximumEntropy

Algorithm

πi queue lengthdistributionLi mean queuelengthXi nodethroughputRi node meanresponse time

Fair for alltheperformanceindices

Fair

Observations

Algorithm for closed QNB: comparison



The approximation principle is network decomposition for all the algorithmsOne-node subneworks as

M/Cox/1/B queue by Tandem Phase-Type DecompositionM/M/1/B queue by the other algorithms

Last algorithm applies the maximum entropy principle

Method Network Constraints BlockingTopology Node type Type

TandemExponential

NetworkDecomposition[Dallery-Fre in’93]

tandem G/M/1/B BAS

Tandem Phase-Type Net wo rk

Decompos ition[Per ros- Altiok’86]

tandem G/M/1/B BAS

Acyc lic Net wo rkDecompos ition

[Lee-a lt.’95]

ac ycli c G/M/1/B BAS

Maximum En trop yAlgo rithm

[Kouva tsos- Xen ios’89]

gene ral G/GE/1/B RS-RD

Methods

Approximate methods for open QNB



BAS blockingExponential service timesPerformance index: network throughput

Tandem Exponential Decomposition and Tandem Phase-Type Decomposition apply network decomposition

M one-node subnetwork T(i), 1≤i≤MT(i) corresponds to node i

analysis of isolated subnetworksT(i) as a M/M/1/Bi queue TEDT(i) as a M/PHn/1/Bi queue TPDefficient solution methods

i

T(i)

1 2 M…

n1 ≤ B1 n2 ≤ B2 nM ≤ BM

Methods

Tandem Networks



All algorithms evaluate for each node iπi queue length distributionLi mean queue length, Xi node throughput, Ri node mean response time

Method Accuracy EfficiencyTandem

ExponentialNetwork

Decomposition

Very goodfor allperformanceindices

Very good

TandemPhase-Type

NetworkDecomposition


Slow when appliedto networks whereall the nodes havefinite capacity andfair otherwise

AcyclicDecomposition


Very good

MaximumEntropy

Algorithm

Good for allperformanceindices

Fair

Observations

Algorithm for open QNB: comparison



most of the equivalencies derive from the identity of the network processes π Remark even if two networks have identical Markov processes, the meaning of corresponding states may be different performance measures may be NOT equivalentequivalence in terms of π does NOT necessarily lead to equivalence in terms of average performance indices

extension of efficient computational algorithms (MVA and Convolution) and solution methods to QNB (e.g.: aggregation technique)

QNB: equivalence Propertiesequivalencies in terms of

- state probability distribution π - average performance indices - passage time distribution

equivalence between networks with and without blocking with different blocking types homogeneous and non-homogeneous networks



Equivalence between networks with and without blocking

Network withRS-RD blocking

Network without blockingparameters

reversible routingsolution PF4

π ∝π* µi*=µifi*(k)=fi(k)/bi(k1),1≤k≤BiP*=P

π∝π* µi*=µihifi*(k)=1/bi(k1),1≤k≤BiP*=P

abitayoutingsoutionPF2

π∝1/π*

µi*=μa jµjfi*(k)=bi(k1),1≤k≤BiP*=||π*ij||,π*ij=µjπji/µi*i≠j,π*ii=1j≠iπ*ji,1≤i,j≤M

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=y A, A=||aij||, aij=pji , j≠i, aii=1-j≠i aij 1≤i,j≤M

(IV)



Equivalence between networks with different blocking types

X and Y blocking typesX=Y identity identical πXY 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

networktopology

Performanceindex

blocking types assumptions

BBS-SO=BBS-ORS-RD =RS-FD

π BBS-SO=RS-RD=RS-FD=BBS-O (I)two- BBS-SO=BBS-SNO (I) and N≤B1+B2-2node BBS-SO →BA (I)anwith

BiBBO=Bi

BA+1π

Ui,Xi,Li,TiBBO=BBO=RR D (II)

πUi,Xi

BBO→BA (II)anwith

BiBBO=Bi

BA+1



Equivalence between networks with different blocking types: closed QNBNetworktopology

perf.index


π BBS-SO=BBS-ORS-RD =RS-FD

π BBS-SO=RS-RD (II)

cyclic

Ui,XiLi,Ti

BBS-SO=BBS-SNO (II), M>2 andN≤min{Bi+Bj:pij>0}-1

πUi, Xi

BBS-SO →BA (II)anBiBB

O=BiBA+1

centa

πUi,XiLi,Ti

BBO=BBNO=BBO=RR D=RFD

(II)anB1<∞,Bi=∞,2≤i≤M

seve π BBO==R RD ==BBNO=BBO→BA

(II)anB1=∞(II),B1=∞an

BiBBO=BiBA+12≤i≤M



Equivalence between networks with different blocking types: open QNB

networktopology

perf.index


BBS-SO=BBS-ORS-RD=RS-FD

tandem π BBS-SO=RS-RD=RS-FDBBS-SO=BBS-SNOBBS-SO→BA

(II)(II),M=2anB1=∞(II)anwithBiBBO=BiBA+12≤i≤M

sπit π BBO=RRD=RFDBBO=BBNO=RFD

(II),B1<∞,Bi=∞,2≤i≤M(II)anB1=∞

BBO=BBOR RD=RFD

μ ege π BBO=RRD=RFDBBO=RRD=RFD==BBNO=BBO

(II)anB1=∞(II)B1<∞,Bi=∞,2≤i≤M

networktopology

perf.index


BBS-SO=RS-FD (II)arbitraryrouting

π BBS-SO=RS-RD=RS-FD=BBS-O (II) and (B)

BBS-SO=BBS-SNO (II) andN ≤ min {Bi+Bj: pij>0}-1



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



Application example of QNBStore-and-forward packet switching networksCircuit 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



Application example of QNB: communication networks

A model of a store-and-forward packet switching network with virtual circuitslevel 3 in OSI reference modelIndependence assumptionsWindow flow control

Closed cyclic network, RS blocking Stations and network nodes have finite bufferPerformance indices network throughput, delay, buffer occupancy

N packetswindow size



N3

N4

C2 D2C1 D1

N1

N2

Computer System Computer System

Communication SubnetworkC1, C2 computer CPU subsystem RS-RD blockingD1, D2 computer Disk subsystem BAS blockingN1, N3 computer network access BAS blockingN2, N4 communication links BBS blocking

Customers represent jobs (in Computer Systems) and packets (in Communication Subnetwork)

Under exponential assumption: heterogeneous QNB reducible to homogeneous QNB RS-RD

solution algorithm: - approximate Maximum Entropy Algorithm- if D1, D2 have RS-RD blocking -> product-form solution F2 - convolution algorithm

Example of heterogeneous QNB: computer-communication system



Queueing Network models with finite capacity queues and blocking can model systems with finite capacity resources and population constraints

QNB are difficult to analyze

Various exact and approximate algorithms Markov process analysis product form solution various approximation with different

efficiency accuracy model constraints and parameters

Heterogeneous networks equivalence and reducibility properties

few algorithms

Open problems algorithms for general heterogeneous QNB multiclass efficient solution tools

Conclusions and open research



Questions?

For further information

S. Balsamo, V. De Nitto Personè, R. Onvural“Analysis of Queueing Networks with Blocking”, Kluwer, 2001

S.Balsamo, D. Kouvatsos Special Issue "Queueing Networks with Blocking"

Performance Evaluation Journal, 2003, 51/2-4

References



BooksS.Balsamo, V. De Nitto Personè, R. Onvural Analysis of Queueing Networks with Blocking. Kluwer Academic Publishers, 2001.Perros, H.G. Queueing networks with blocking. Oxford University Press, 1994.Special IssuesS.Balsamo, D. Kouvatsos Special Issue "Queueing Networks with Blocking”, Performance Evaluation Journal, 2003, 51/2-4.Onvural, R.O. Special Issue on Queueing Networks with Finite Capacity, Performance Evaluation, Vol. 17, 3 (1993).Akyildiz, I.F., and H.G. Perros, Special Issue on Queueing Networks with Finite Capacity Queues, Performance Evaluation, 10/3 ,1989.Survey papersS.Balsamo, V.De Nitto "A survey of Product-form Queueing Networks with Blocking and their Equivalences" Annals of Operations Research, vol. 48, Jan 1994.Onvural, R.O. "Survey of Closed Queueing Networks with Blocking" ACM Computing Surveys, Vol. 22, 2 (1990) 83-121.Perros, H.G. "Open Queueing Networks with Blocking" in Stochastic Analysis of Computer and Communications Systems, (Takagi Ed.), Elsevier Science Publishers, North Holland, 1989.PapersAkyildiz, I.F. "Exact Product Form Solutions for Queueing Networks with Blocking" IEEE Trans. on Computers, Vol. 1 (1987) 121-126.Akyildiz, I.F. "On the Exact and Approximate Throughput Analysis of Closed Queueing Networks with Blocking" IEEE Trans. on Software Eng., Vol. 14 (1988), 62-71.Akyildiz, .F. Mean value analysis of blocking queueing networks, IEEE Trans. Soft. Eng. 14 (1988) 418–429.Akyildiz, I.F., and S. Kundu "Deadlock Free Buffer Allocation in Closed Queueing Networks" Queueing Systems Journal, 4 (1989) 47-56.Akyildiz, I.F., and H. Von Brand "Exact solutions for open, closed and mixed queueing networks with rejection blocking" J. Theor. Computer Science, 64 (1989) 203-219.

References 1



Akyildiz, I.F., and N. Van Dijk "Exact Solution for Networks of Parallel Queues with Finite Buffers" in Performance ’90 (P.J.B. King, I. Mitrani and R.J. Pooley Eds.) North-Holland, 1990, 35-49.Altiok, T. and H.G. Perros, Approximate analysis of arbitrary configurations of queueing networks with blocking, Ann. Oper. Res. 9 (1987) 481-509.Ammar, M.H., and S.B. Gershwin "Equivalence Relations in Queueing Models of Fork/Join Networks with Blocking" Performance Evaluation, Vol. 10 (1989) 233-245.Awan, I.U. and D.D. Kouvatsos, Approximate analysis of QNMs with space and service priorities, in: D.D. Kouvatsos (Ed.), Performance Analysis of ATM Networks, Kluwer Academic Publishers, IFIP Publication, Chapter 25, 1999, pp. 497–521.Balsamo, S."Closed Queueing Networks with Finite Capacity Queues: Approximate analysis" Proc. ESM'2000, SCS, European Simulation Multiconference 2000, Ghent, 23-26 May 2000.Balsamo, S., C. Clò "A Convolution Algorithm for Product Form Queueing Networks with Blocking" Annals of Operations Research, Vol. 79 (1998) 97-117.Balsamo, S., M.C. Clo' L.Donatiello "Cycle Time Distribution of Cyclic Queueing Network with Blocking" Performance Evaluation, North Holland, vol.14, n.3, 1993.Balsamo, S., V. De Nitto Personè, P.Inverardi "A review on queueing network models with finite capacity queues for software architectures performance prediction" Performance Evaluation Journal, 2002, 51/2-3 pp. 269-288. Balsamo, S., and L. Donatiello "On the Cycle Time Distribution in a Two-stage Queueing Network with Blocking" IEEE Transactions on Software Engineering, Vol. 13 (1989) 1206-1216.Balsamo, S., L. Donatiello and N. Van Dijk “Bounded performance analysis of parallel processing systems” IEEE Trans. on Par. and Distr. Systems, Vol. 9 (1998) 1041-1056.Balsamo, S., and G. Iazeolla "Some Equivalence Properties for Queueing Networks with and without Blocking" in Performance '83 (A.K. Agrawala, S.K. Tripathi Eds.) North Holland, 1983.Baskett, F., K.M. Chandy, R.R. Muntz, and G. Palacios "Open, closed, and mixed networks of queues with different classes of customers" J. of ACM, 22 (1975) 248-260.Boucherie, R. "Norton's Equivalent for queueing networks comprised of quasireversible components linked by state-dependent routing" Performance Evaluation, Vol. 32 (1998) 83-99.

References 2



Boucherie, R., and N. Van Dijk "On the arrival theorem for product form queueing networks with blocking" Performance Evaluation, 29 (1997) 155-176.Bouchouch, A., Y. Frein and Y. Dallery "Performance evaluation of closed tandem queueing networks with finite buffers" Performance Evaluation, Vol. 26 (1996) 115-132.Boxma, O., and A.G. Konheim "Approximate analysis of exponential queueing systems with blocking" Acta Informatica, 15 (1981) 19-66.Brandwajn, A., and Y.L. Jow "An approximation method for tandem queueing systems with blocking" Operations Research, Vol. 1 (1988) 73-83.Buzacott, J..A., and J.G. Shanthikumar "Design of Manufacturing Systems using Queueing Models" Queueing Systems: Theory and Applications, (1992).Caseau, P., and G. Pujolle "Throughput capacity of a sequence of transfer lines with blocking due to finite waiting room" IEEE Trans. on Softw. Eng. 5 (1979) 631-642.Cheng, D.W. "Analysis of a tandem queue with state dependent general blocking: a GSMP perspective" Performance Evaluation, Vol. 17 (1993) 169-173.Clò, C. "MVA for Product-Form Cyclic Queueing Networks with RS Blocking" Annals of Operations Research, Vol. 79 (1998).Dallery, Y., and Y. Frein "On decomposition methods for tandem queueing networks with blocking" Operations Research, Vol. 14 (1993) 386-399. Dallery, Y., Z. Liu, and D.F. Towsley "Equivalence, reversibility, symmetry and concavity properties in fork/join queueing networks with blocking" J. of the ACM, Vol. 41 (1994) 903-942.Dallery, Y., and D.F. Towsley "Symmetry property of the throughput in closed tandem queueing networks with finite buffers" Op. Res. Letters, Vol. 10 (1991) 541-547.De Nitto Personè, V. "Topology related index for performance comparison of blocking symmetrical networks" European J. of Oper. Res., Vol. 78 (1994) 413-425.Frein, Y., and Y. Dallery "Analysis of Cyclic Queueing Networks with Finite Buffers and Blocking Before Service", Performance Evaluation, Vol. 10 (1989) 197-210.Gavish, B., and I. Neuman "Capacity and Flow Assignments in Large Computer Networks" in Proc. IEEE-Infocom'86, 1986, 275-284.Gershwin, S. B. "An efficient decomposition method for the approximate evaluation of tandem queues with finite storage space and blocking" Oper. Res., Vol. 35 (1987) 291-305.

References 3



Gordon, W.J., and G.F. Newell "Cyclic queueing systems with restricted queues" Oper. Res., Vol. 15 (1967) 286-302.Gün, L., and A.M. Makowski "An approximation method for general tandem queueing systems subject to blocking" Proc. First Int. Workshop on Queueing Networks with Blocking, (H.G. Perros and T. Altiok Eds.) North Holland, 1989,147-171.Hillier, F.S., and W. Boling "The Effect of Some Design Factors on the Efficiency of Production Lines with Variable Operation Times" J. Ind. Eng., Vol. 7 (1966) 651-658.Hillier, F.S., and W. Boling "Finite queues in series with exponential or Erlang service times - a numerical approach" Oper. Res., Vol. 15 (1967) 286-303.Hillier, F.S., and K.C. So "The assignment of extra servers to stations in tandem queueing systems with small or no buffers" Performance Evaluation, Vol. 10 (1989) 213-231.Hordijk, A., and N. Van Dijk "Networks of queues with blocking", in: Performance '81 (K.J. Kylstra Ed.) North Holland (1981) 51-65.Jafari, M. A. and J.G. Shanthikumar "Determination of Optimal Buffer Storage Capacities and Optimal Allocation in Multistage Automatic Transfer Lines" IIE Trans., Vol. 21 (1989) 130-135.Jun, K.P., and H.G. Perros "An approximate analysis of open tandem queueing networks with blocking and general service times" Europ. Journal of Operations Research, Vol. 46 (1990) 123-135.Kelly, K.P. Reversibility and Stochastic Networks. J. Wiley and Sons Ltd., Chichester, England, 1979.Konhein, A.G., and M. Reiser "A queueing model with finite waiting room and blocking" SIAM J. of Comput, Vol. 7 (1978) 210-229.Kouvatsos, D.D. "Maximum Entropy Methods for General Queueing Networks" in Proc. Modeling Tech. and Tools for Perf. Analysis, (Potier Ed.), North Holland, 1983, 589-608. Kouvatsos, D., and I.U. Awan "Arbitrary closed queueing networks with blocking and multiple job classes" Proc. Third International Workshop on Queueing Networks with Finite Capacity, Bradford, UK, 6-7 July, 1995.Kouvatsos, D.D. and I.U. Awan, MEM for arbitrary closed queueing networks with R–S blocking and multiple job classes, Special Issue on Queueing Networks with Blocking, vol. 79, Baltzer Science Publishers, 1998, pp. 231–269. Kouvatsos, D., and S.G. Denazis "Entropy maximized queueing networks with blocking and multiple job classes" Performance Evaluation, Vol. 17 (1993) 189-205.

References 4



Kouvatsos, D.D. and N.P. Xenios, MEM for arbitrary queueing networks with multiple general servers and repetitive-service blocking, Perform. Eval. 10 (1989) 106–195. Kouvatsos, D.D., and N.P. Xenios "Maximum Entropy Analysis of General Queueing Networks with Blocking", in First Int. Work. on Queueing Networks with Blocking, (Perros and Altiok Eds), Elsevier Science Publishers North Holland, 1989.Kundu, S., and I. Akyildiz "Deadlock free buffer allocation in closed queueing networks" Queueing Systems Journal, Vol. 4 (1989) 47-56.Lam, S.S. "Queueing networks with capacity constraints" IBM J. Res. Dev., Vol. 21 (1977) 370-378.Lee, H.S., and S. M. Pollock "Approximation analysis of open acyclic exponential queueing networks with blocking" Operations Research, Vol. 38 (1990) 1123-1134.Lee, H.S., A. Bouhchouch, Y. Dallery and Y. Frein "Performance Evaluation of open queueing networks with arbitrary configurations and finite buffers" Proc. Third Int. Work. on Queueing Networks with Finite Capacity, Bradford, UK, 6-7 July, 1995.Liu, X.G., and J.A. Buzacott "A decomposition related throughput property of tandem queueing networks with blocking" Queueing Systems, Vol. 13 (1993) 361-383.C. Lladò, P. Harrison, A new blocking problem from Java-based schedulers implementation, Performance Evaluation, 51/2-4, Feb. 2003, 229-246.Mishra, S., and S.C. Fang "A maximum entropy optimization approach to tandem queues with generalized blocking" Performance Evaluation, Vol. 30 (1997) 217-241.Mitra, D., and I. Mitrani " Analysis of a Kanban discipline for cell coordination in production lines I" Management Science, Vol. 36 (1990) 1548-1566.Mitra, D., and I. Mitrani "Analysis of a Kanban discipline for cell coordination in production lines II: Stochastic demands" Operations Research, Vol. 36 (1992) 807-823.Neuts, M.F. "Two queues in series with a finite intermediate waiting room" J. Appl. Prob., 5 (1986) 123-142.Onvural, R.O. "Some Product Form Solutions of Multi-Class Queueing Networks with Blocking' Performance Evaluation, Special Issue on Queueing Networks with Blocking, (Akyildiz and Perros Eds), 1989.

References 5



Onvural, R.O., and H.G. Perros "On Equivalencies of Blocking Mechanisms in Queueing Networks with Blocking" Oper. Res. Letters, Vol. 5 (1986) 293-298.Onvural, R.O., and H.G. Perros "Equivalencies Between Open and Closed Queueing Networks with Finite Buffers" Performance Evaluation, 1988.Onvural, R.O., and H.G. Perros "Some equivalencies on closed exponential queueing networks with blocking" Performance Evaluation, Vol.9 (1989) 111-118.Onvural, R.O., and H.G. Perros "Throughput Analysis in Cyclic Queueing Networks with Blocking" IEEE Trans. Software Engineering, Vol. 15 (1989) 800-808. Perros, H.G., and T. Altiok "Approximate analysis of open networks of queues with blocking: tandem configurations" IEEE Trans. on Software Eng., Vol. 12 (1986) 450-461.Perros, H.G., A. Nilsson, and Y.C. Liu "Approximate Analysis of Product Form Type Queueing Networks with Blocking and Deadlock" Performance Evaluation (1989).Perros, H.G., and P.M. Snyder "A computationally efficient approximation algorithm for analyzing queueing networks with blocking" Performance Evaluation, Vol. 9 (1988/89) 217-224.Ramesh, S. and H. Perros, A multilayer client–server queueing network model with synchronous and asynchronous messages, IEEE Trans. Soft. Eng. 26 (11) (2000) 1086–1100. Reiser, M. "A Queueing Network Analysis of Computer Communications Networks with Window Flow Control" IEEE Trans. on Comm., Vol. 27 (1979) 1199-1209.Sereno, M. "Mean Value Analysis of product form solution queueing networks with repetitive service blocking" Performance Evaluation, Vol. 36-37 (1999) 19-33.Shanthikumar, G.J., and D.D. Yao "Monotonicity Properties in Cyclic Queueing Networks with Finite Buffers" in First Int. Work. on Queueing Networks with Blocking, (Perros and Altiok Eds), Elsevier Sci. Pub., North Holland, 1989.Suri, R. and G.W. Diehl, A variable buffer size model and its use in analytical closed queueing networks with blocking, Management Sci. Vol.32, 2 (1986) 206-225. Van Dijk, N. "Queueing networks and product form" John Wiley (1993).Van Dijk, N., E. van der Sluis, Simple Product-form bounds for queueing networks with finite clusters" Annals of op. Res. 113, -4, Jul. 2002.Yao, D.D., and J.A. Buzacott "Modeling a Class of State Dependent Routing in Flexible Manufacturing Systems" Annals of Operations Research, Vol. 3 (1985) 153-167.

References 6



Convolution algorithm for product-form closed QNB

Details of the approximation algorithms for closed QNBopen QNB

Observations on method comparison

Details on product-form conditions

Additional method information



Convolution algorithm for QNB

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Computation of values aj, Aj, B(j)

Initialisation

Computation of functions Gj(n)

for j=1 to M {

}for n =a1 to B1

;for j=2 to M {MIN = min (Aj-1+Bj, B(j-1)+aj);MAX = max (Aj-1+Bj, B(j-1)+aj;);

;for n= Aj +1 to min(MIN, N)

;for n= MIN +1 to min(MAX ,N)if MIN = B(j-1)+aj

then ;

else ;for n=MAX +1 to min(B(j)-1, N)

;if N > B(j) then

}

€

a j = max(0, N − Bk1≤k≤M, k≠j

∑ )

€

Aj = a i1≤i≤j∑;

€

B(j) = Bi1≤i≤j∑;

€

G1(n) = ρ 1n

€

Gj (Aj ) = ρja j

Gj−1(Aj−1)

€

Gj (n) = ρj

a jGj−1(n − a j ) + ρj Gj (n − 1)

€

Gj n( ) = ρj Gj n−1( )

€

Gj (n) = ρja j

Gj−1(n − a j ) − ρjBj+1

Gj−1(n − Bj − 1) + ρjGj (n − 1)

€

Gj n( ) = − ρjBj+1

Gj−1 n−Bj −1( ) + ρj Gj n−1( )

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Gj (B(j)) = ρjBj

Gj−1(B(j - 1))



BBS or BAS blockingAssumptionthroughput is a symmetrical function of N X(N)=X(B-N)holds for BBS blocking as proved for phase-type service distributions [Dallery-Towsley ’91]conjecture for BASBBS : maximum throughput for N=N*N*=B/2B even

N*=B/2,B/2+1B odd

Exact computation of few values of X(N)Interpolation by

X(N)=X(N+1)- y xN*-N

where y=[X(N*) -X(B-)]/ 1≤i≤(N*-B -) xi

and x is the fixed-point of

X(B--1)=X(B-)-[X(N*)-X(B-)][xN*-B-+1(1-x)]/[x-xN *-B -+1]

B- = min 1≤i≤N Bi

1 N* B

non-decreasing non-increasing

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Cyclic Networks: Throughput Approximation



BAS blockingconjectured maximum throughput for N =N* dependent on queue capacities and service ratesN* i(Bi+1)/2 - 1More evaluation of X(N)Approximation for N*≤N≤B-2Computational complexity >> exact analysis

1. Exact computation of X(N) for N=B--1, B-, N*. for 1≤N≤B- network without blocking: product-form algorithm X(N*) solution of the associated Markov chain

2. Approximation of X(N) for B-+1≤N≤N*-1. formulas and solution of the fixed-point problem

For BAS blocking two additional steps:3. Exact computation of X(N) for N=B-1, B.

• X(B) as the average time between two successive deadlocks which are immediately detected and resolved: numerical integration; • X(B-1) approximated by a function of X(B) or directly computed

4. Approximate computation of X(N) for N*+1≤N≤B-2, as at step 2.

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Cyclic Networks: Throughput Approximation



BBS blocking1.network decomposition: M one-node subnetworks2. analysis of isolated subnetworks

M/M/1/Bi arrival rate i* and load dependent service rate µ*i(n), 0≤n≤Bi to derive the marginal queue length distribution p*i (n), 0≤n≤Bi 1≤i≤M

3. approximate aggregation

• CASE B1=∞ µ*i(n)={(1/µ i) + 1≤I≤Mb ij(n)[i+1≤k≤j (1/µ k)]}-1

µ*M(n)=µM n ,i i *= X/(1-p*i(Bi)) (*)

X network throughputb ij(n) probability that nodes i+2,…,j are full and node j+1 is not full, given n

customers in node i, 1≤i,j≤M b ij(n) in terms of p*k (n), I+1≤k≤M

p*i(Bi) function of i *Given X, i * is the fixed point of equation (*)Iterative scheme

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Cyclic Networks - Network Decomposition



Iterative scheme • CASE B1= ∞

• CASE B1<∞

Approximation of µ*M(n)=µ and an additional iteration cycle to compute p*i(Bi),i Computational complexity O( k M4(B+)3) operations for k iteration steps, B+=max iBi

Input: approximate throughput [Xmin(0), Xmax(0)]

Repeat

(step k≥1)

• computes new parameters i* and µ*i(n), 0≤n≤Bi, 1≤i≤M

• appropriately updates the k-th throughput approximation [Xmin(k), Xmax(k)]

Until [(Xmax(k)- Xmin(k))<eand

average nodes population N and

i*<µ*i-1,i ]Output: approximate throughput

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BBS blocking and assume B1=∞ network decomposition applied to nested subnetworks

Ci has with load dependent service rate ni(n) and a variable queue capacity

fi(n|N) fraction of time in which the queue capacity is n, given N customers in the network 1≤n≤N

Analysis of two-node subnetworks with a composite node with variable queue capacity (variable buffer) (VB)

consider two corresponding two-node networks with a composite node with fixed buffer (FB) and with infinite buffer (IB), respectively

FB and IB have a simple closed-form solution

i

i i+1 i+2 M

Ci

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Parameters ni(n) and fi(n|N) of each VB two-node network are derived by the solutionof the two corresponding FB and IB networks.VB network parameters and state probabilities are defined as a weighted sum of the FB parameters and state probabilities; these are in turn approximated by using theIB model solution.

Simple, non iterative algorithm Computational complexity O(MN3) operations

• Analysis of subnetwork {M-1,M} to define the aggregate node CM-1, seen by node M-2

• From node i=M-2 to node 1 analysis of subnetwork {i, Ci } to define the aggregate node Ci-1

• At the last step the network {1, C1}represents the entire aggregated network

Obtain the approximated throughput

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BAS blocking and exponential service timeapproximate the network with blocking with a network without blocking by choosing N to approximately match the state space cardinality of the underlying Markov chainAssumptionthe two networks with nearly the same state space cardinality should have similar throughputs

K(N) state space cardinality of the Markov chain associated to the network with blocking with N customers

K'(N') state space cardinality of the Markov chain associated to the network without blocking with N' customers

Determine N' to approximate K(N)=K'(N') Analyze the network without blocking

Simple implementation Computational complexity O(M3+MN2)

1. Compute K(N) by a convolution algorithm2. Determine N' to minimize |K(N)-K'(N')|, 1≤N'≤N, by linear search in[1,N]3. Compute the throughput of the network without blocking by a

convolution algorithm

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Arbitrary Topology Networks:Matching State Space



BAS blocking and exponential service timeModified MVA algorithm defined for product-form networks with unlimited queue capacities and based on

Little lawarrival theorem which that does not hold for Q.N. with blocking

Recursive schemeRi(n)= (1/µi) [1+Li(n-1)] 1≤i≤M (1)Xi(n)= n ei/[1≤j≤M ejRj (n)] 1≤i≤M (2)Li(n)= Xi (n) / Ri (n) 1≤i≤M (3)

Simple implementation Computational complexity O(M3+ k MN )k iteration number of the internal cycle at step 2

1. Initialization2. For each population n=1,…N

Repeat computation of MVA equations (1)-(3) where (1) is substituted byRi(n)= (1/µi) Li(n-1) for a full node i

Rj(n)= (1/µj) Lj(n-1 )+ (1/µi) (ejpji /ei) for a blocked node j

until Li(n)≤Bi for each node i.

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Arbitrary Topology Networks- Approximate MVA



RS-RD blocking and generalized exponential service timeMaximum entropy principleApproximation of π(S) for each network state S=(n1,…, nM) by maximizing the entropy function

H(π)=-S π(S)log(π(S)) subject to(I) (normalization) S π(S)=1

(II) (probability of ni≥ ai) ni>aiπi(ni)=ui

(III) (mean queue length) ai≤ni≤Bihi(ni)πi(ni)=Li

(IV) (full node) ai≤ni≤Bifi(ni)πi(ni)=Fi

ai=max{0, N-B+Bi} minimum node i populationhi(ni)=min{0, ni-ai-1} and f(ni)=max{0, ni-Bi+1}

Product form approximation by the Lagrange's method of undetermined multipliers

π(n)=(1/Z) P1≤ i ≤M xi(ni) yihi(ni ) zi

fi

Z normalizing constant,xi(ni)=1if ni=ai, xi(ni)=xi if ai≤ni≤Bi,xi, yi and zi are the Lagrangian coefficients corresponding to constraints (II)-(IV)

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Arbitrary Topology Networks: Maximum Entropy Algorithm



The network cannot be decomposed into single nodes and coefficients xi, yi and zi do not have a closed form expressionApproximation of the closed network with a pseudo open without exogenous departures and arrivalsApproximate analysis of the open network

by adding the constraint N=i Li

slight modifications to derive a solution for x i, yi

iterative approximation for zi” Approximate analysis of the open network

Computational complexity: algorithm for step 1 (open network) and O(kM2N2) for step 2

1. Analysis of the pseudo open network with the approximation for open networks slightly modified to derive coefficients xi, yi and an approximation for zi , i

2. Iterative evaluation of coefficients zi by a convolution algorithm to compute network throughputs

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Arbitrary Topology Networks: Maximum Entropy Algorithm



ObservationsCyclic Networks

Network Decomposition (ND) is more accurate than Variable Queue Capacity Decomposition (VQCD) for both the average and the maximum relative error. This difference increases with M.

Throughput Approximation (TA) is more accurate than ND for both the average and the maximum relative error. TA accuracy is more stable that ND as M increases.

ND is more efficient than TA, which is limited to small networksthe time computational complexities of ND (O(kM4(B+)3)) and of VQCD

(O(MN3)) show a different dependence on network parameters. If N<MB+ then VQCD approximation is better than ND, worse otherwise. VQCD approximation is less efficient than the ND for large N.

VQCD and TA provide the throughput for all the network population from 1 to N

fixed point iteration in ND can show some numerical instability (observed for M=20)

ND and TA apply to a more general class than VQCD

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ObservationsArbitrary Topology Networkscompare the two approximation algorithms Matching State Space (MSS) and Approximate MVA (AMVA) that apply to the same networksMSS is more accurate than AMVA both in terms of average and

maximum relative errorsapproximations are quite different, their rationales are not relatedboth MSS and AMVA seem to be independent of network

parameters (M, µi,B i), but dependent on the topology. Better results for central server networks and worse results for cyclic networks

MSS is more efficient than AMVAboth algorithms are stable

The Maximum Entropy Algorithm applies to a more general class of networks

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BAS blocking subsystem T(i): M/M/1/ Bi

µu(1) = ,µd(M) = µM,2 (M-1) unknowns

• pb(i) probability that at arrival time T(i) is full • ps(i) probability that at the end of a service T(i) is empty

µu(i) = [(1/µ i-1)+ps(i-1) / µu(i-1)] -1 2≤i≤M (1)µd(i) = [(1/µi)+pb(i+1) / µd(i+1)] -1 1≤i≤M-1 (2)X1=X2=…=XM

3 equivalent systems to determine the unknowns

Computational complexity O(k M (B+)2)k iteration number at step 2

0. Initialization: µu(1)=, µd(i)=µi i1. Repeat

1a forward cycle: for i=1,…, M-1

compute ps(i) and µu(i+1) by (1)1b backward cycle: for i=M,…, 2

compute pb(i) and µd(i-1) by (2)

until max{|Xi-Xj|, 1≤i,j≤M}<e

2. Compute Li, πi (n), 0≤n≤Bi, 1≤i≤M and X1

ni≤ Bi

T(i)

µu(i)arrival rate service rate

µd(i)

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Tandem Exponential Decomposition



BAS blockingsubsystem T(i): M/ PHn /1/ Bi

T(M) is M/M/1/BM +1 with service rate µM Service with M-i+1 exponential phases to consider blocking due to nodes (i+1,…,M)

PHn is represented by the pair (ai,Ti), ai=[1,0,0,…0] an (M-i+1)-vector

Ti=[µi, µI+1,…,µM]TA, A=[ars] (i≤r,s≤M) upper triangular square matrix of probabilities among the exponential phases

aij=ji+1wi+1(j) 1≤i≤M, i+1≤j≤M (1)ji=πi(Bi+1)/[iwiTi

-1 1] 2≤i≤M-1 (2)wi(j)=πi(0)aiRi

Bi gij)/π i(Bi) 2≤i≤M-1, i+1≤j≤M (3)

wi=[wi(j)] (i+1≤j≤M) row (M-i+1)-vector, 1=[1…1]T

gij column (M-i+1)-vectors with all 0 except 1 in j-th

• fixed point problemi=i/[1-π i(Bi+1)] (4)

T(i)

i

≤ Bi+1

arrival rate Phase-Type service rate

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Tandem Phase-Type Decomposition



If the first node has unlimited capacity (B1=)

If the first node has finite capacity (B1<) , the first arrival rate 1 ≠ exogenous arrival rateAdd an iterative cycle to estimate the effective arrival rates i for each node iConvergence has not been provedComputational complexity O(k12≤I≤M ki(M-i+1)3Bi

2) ki iteration number to compute i, 2≤i≤M-1, at step 2

and external iteration cycle for i=1

1. Analysis of subsystem M • 1a determine M as the fixed point solution of (4)• 1b compute πM by the M/M/1/BM+1 analysis• 1c compute jM-1=µMπM(BM+1)/M

2. Analysis of subsystem T(i), i=M-1,…, 2• 2a determine i as the fixed point solution of (4)• 2b compute πi by the M/PHM-i+1/1/Bi+1 analysis• 2c compute wi(j), ji and ai-1j, j, by (3) (2) and (1)

3. Analysis of subsystem T(1)compute π1 by M/PHM/1/B1+1 analysis with arrival rate

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Tandem Phase-Type Decomposition



BAS blocking, extension of the method of Tandem Exponential DecompositionNetwork decomposition into M one-node subsystems T(i): M/M/1/ Bi 1≤i≤M• Ui={j: pij>0} predecessor of node I

T(i) receives from |Ui| exponential sources with rates µuj(i), for jUi

New set of equations to determine subnetwork parameters, new formulas for unknown rates µuj(i) and µd(i)• pbj(n:i) probability that at arrival time at T(i) from the j-th

source n nodes are blocked by node i, 0≤n<|Ui|, jUi• ps(i) probability that at the end of a service T(i) is empty

µd(i)

T(i)µuj(i)

ni ≤ Biexogenous arrival rate

service rate

i

sending nodes

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Acyclic network decomposition



Computational complexity bounded byO(kM [(U+B+)2+U3+2 U+1]) k iteration number and U=maxi|Ui|

Note that for tandem networks U=1 and O(kM(B+)2) as Tandem Exponential Decomposition algorithm

0. Initialization: µu1(1)=i µd(i)=µi ”I

1. Repeat1a forward cycle: for i=1,…, M

compute ps(i)= p(1:i)/[1-p(0:i)]

and µuj(i) j in U j1b backward cycle: for i=M,…, 1

compute pb(n:i) and µd(i) until convergence of µd(i)

2. Compute Li, Xi , Ri and πi (n), 0≤n≤Bi, 1≤i≤M

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Acyclic network decomposition



RS-RD blocking and generalized exponential service timeMaximum entropy principleNetwork decomposition into M one-node subsystems T(i): GE/GE/1/ B 1≤i≤MApproximation of node i queue length distribution πi(ni) by maximizing the entropy function

H(π i)=-n πi(n) log(πi(n)) subject to

(I) (normalization) 0≤n≤Bi πi(n)=1

(II) (utilization) 0≤n≤Bi hi(n) πi(n)= i

(III) (mean queue length) ai≤n≤Bi nπi(n)=Li

(IV) (full node) ai≤n≤Bi f(ni)πi(ni)=Fi

ai=max{0, N-B+Bi} minimum node i populationhi(ni)=min{1, max(0,n)} and f(ni)=max{0, ni-Bi+1}Product form approximation by the Lagrange's method of undetermined multipliers

π(n)=(1/Z) P1≤ i ≤M xihi(ni ) yi

ni zifi(ni )

Z normalizing constant,xi=e-bi1 , yi=e-bi2 , zi=e-bi3

bij are the 3M Lagrangian coefficients corresponding to constraints (II)-(IV)

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Maximum Entropy Algorithm



Network decomposition into single nodesThe marginal queue length distribution for node i reduces to

πi(n)=(1/Z') xihi(ni ) yi

ni zifi(ni )

Z' normalizing constant

GE/GE/1/B with arrival rate i and service rate µ’icdi coefficient of variation of the interarrival time

csi coefficient of variation of interdeparture time

Computational complexity O() cardinality of the set of probabilities computed at step 1b

Initialization (including cdi i)1. Repeat

for each subsystem T(i), 1≤i≤M1a compute the effective arrival rate 'i by the traffic equations1b compute the probability that at service completion time at i node j is full, j1c compute queue length probability πi 1d compute new values for cdi

until convergence of cdi

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Maximum Entropy Algorithm



Algorithm for open QNB: comparisonObservations• Tandem Networks BAS exponential networks: comparison of the two approximations by Tandem Exponential Decomposition and Tandem Phase-Type Decomposition

the accuracy of the two methods is almost similar. TPD is slightly better than TED for high blocking probabilities

the two approximations are quite similar, for sign and valueaccuracy of TPD is influenced by capacity queue (B) unbalancing, while TED

is affected by service rate (µ) unbalancingboth TPD and TED accuracy increases for small blocking probabilities, i.e.

for networks with B>> or µ/ >> TED is certainly more efficient than TPDTPD can show numerical instability that can affect the algorithm convergenceimplementation of TED is simpler than TPD

• The Maximum Entropy Algorithm applies to a more general class of QNB with RS blocking

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Numerical comparison of approximated methods

QNBA Queueing Networks with Blocking Analyzerspecification and analysis of queueing networks with finite capacity queues and population constraints

User Interface

Model Analysis

Preliminaly

Analysis

Exact Analysis

Computation of

performance indices

Results

Model Analysis

Approximate Analysis

Results

Algorithm Selection

Open Networks

... ...

Module

Algorithm 1

Module

Algorithm 4

Module

Algorithm 5

Module

Algorithm

10

Model Solution

Algorithm Selection

Closed Networks

Produc-form

Algorithms

Model Solution

Markov chain

generation

Markov chain

solution

Analysis of QNB



Comparison with exact or simulation resultsRelative error on performance measures

REL=[(A-B)/A]100% percentage relative error of the approximate result (A) and the exact or simulation result (B) defined as follows

A-REL average of percentage relative errorsMax-REL maximum of the percentage relative errors

Parametric analysismodel parameters

nodes, customers, topology, service rates, queue capacitySymmetry of network parameters

Service rate unbalancingQueue capacity unbalancing

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Numerical Results



Condition 1: non-empty condition for closed networks at most one node can be emptyN≥B- B-

Condition 2: strictly non-empty condition each node can never be empty N>B- B-

B= i Bi B- = min i Bi

Condition 3: for a particular model of multiclass networks with parallel queues with interdependent blocking functions and service rates, and which satisfy a so-called invariant condition [AV90]

Condition 4: each node i with finite capacity is the only destinationnode for each upstream node if pji > 0 then pji = 1, 1≤j≤MCondition 5: only one node blockedat most one node can be blocked if N=B- +1

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Product-form conditions

Queueing Networks with Blocking analysis, algorithms and properties

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