Queueing Networks with Blocking

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

I)
II)
III)
IV)
3
Queueing networks with blocking (QNB)finite
capacity queues
(I)

• 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 ? 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
Queueing networks with blocking (QNB)finite
population constraint

• (sub)network population constraint
• n number of customers in the network
• B network finite capacity
• if nB 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
• BAS Blocking After Service
• BBS Blocking Before Service
• RS Repetitive Service Blocking
• Queueing networks with (sub)network population
  constraint
• STOP
• Recirculate

6
Blocking Types

• QNB with finite capacity queues
• BAS Blocking After Service
• BBS Blocking Before Service
• RS 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 scheduling
• First Blocked First Unblocked

7
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 repeat the service in the
  sending node
• RS-RD vs RS-FD (Random or Fixed Destination)

8
Blocking Types

• QNB with population costraints
• n?L,U
• a(n) 0 for nU load dependend arrival rate
• d(n) 0 for nL 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, 1i,jM, n0.

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
lt
population
10
QNB performance indices

• Related to a single resource i (a service center)
  
• average indices
• random variables
• Ni number of customers in the resource
• ti customer passage time through the resource
• distribution of ni
• pi(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 nodes l total arrival rate
• N number of customers (closed network) µi service
  rate of node i
• Ppij routing matrix p0i arrival
  probability at node i
• xi visit ratio at node i, solution of traffic
  equations
• Bi finite capacity of node i bi(ni) blocking
  function
• 0ltbi(ni)1, for 0niltBi, bi(Bi)0
• Performance indices
• depend on the blocking type
• are derived from the state probability pi(ni)
  or ? (ni)

xi l p0i ?j xj pji
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Performance indices

• For single server node i
• utilization Ui 1 - pi(0) - PBi
• throughput Xi ?ni pi(ni) -
  PBi(ni) µi(ni)
• Xi Ui µi for constant service rate
• mean queue length Li ?ni ni pi(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 ?ni PBi(ni) overall blocking probability
• PB definition depends on the blocking type
• Effective utilization when the server are neither
  empty nor blocked
• Effective throughput the useful work (for RS and
  BBS)

13
Analytical solutions for QNB

• Evaluation of average performance indices and
  joint queue length distribution at arbitrary
  times (p)
• exact solution
• based of Markov process analysis
• product-form solution of p
• approximate and bound solution

14
Analytical solutions for QNB
(II)

• S (S1,,SM) system state
• Si node i state which includes ni , 1iM
• E set of all feasible states
• E discrete state space
• Q infinitesimal generator
• if P (network routing matrix) irreducible
• then ? ! stationary state distribution p
  p(S), S?E
• solution of the global balance equations
• the definition of S, E and Q depends on

Markovian network the network behavior can be
represented by a homogeneous continuous time
Markov process M
p Q 0 , ?S?E p(S) 1
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A simple example two-node cyclic network

• FCFS service discipline
• exponential service time
• S (S1,S2) system state definition
• Si ni RS or BBS blocking
• if ni Bi RS server active, BBS server
  blocked
• Si (ni, si) BAS blocking
• where si is the server state si1 (active)
  si0 (blocked)
• birth-death Markov process
• closed-form solution

16
A simple example two-node cyclic network

• Let ? (µ1/µ2)
• for BBS and RS
• p(S)(1/C) ? n2-NB1 ?S (n1, n2 )?? E
  CS 0iB1B2-N ?i
• for BAS
• p(S)(1/C) ? n2-NB11 S((n1,1),(n2,1)) ? E
• p(S)(1/C) S((B1,1),(N-B1,0))
• p(S)(1/C) ? B2B12-N S((N-B2,0),(B2,1))
• C S 0iB1B22-N ?i
• for infinite capacity queues (no blocking)
• p(n1, n2)(1/C) ? n2 0n1N , n2N-n1
  C S 0iB1B22-N ?i

17
Analytical solutions for QNB state definition

• S(S1,,SM) system state
• Si state of node i which includes ni, 1iM
• exponential network, First Come First Served
  discipline, general topology
• di the destination node of the next job that
  will exit from node i
• si server state active (1) blocked (0)
• mi (mi,,mu(i)), 0u(i)M-1
• queue of indices of the nodes blocked by node i,
  if niBi
• unblocking scheduling
• RS the server is always active
• BBS-SO the server is blocked if nigt0 and
  ndiBdi
• BBS-SNO idem and niltBi

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 ni0, ?(ni)1 otherwise, 1iM
• ei M-vector with all zero except one in i-th
  position
• q(S,S) - ?S'?E, S'?S q(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 (p) 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 p
4 Computation from p 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
• approximate solution methods

20
Exact analysis of QNB special cases

• subset of Markovian networks
• product-form solution of p
• 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

21
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, pj11 for
2jM,
?ni , 1iM
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 ? xi pij xj pji ?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 which the throughput of the network
  is shown to be symmetric with respect to its
  population
• BSi Bi X(N-B) X(B)

26
Product-form QNB algorithms

• Algorithms for closed QNB
• 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, arbitrary topology, load
  independent service rates
• F1 or F2 product form solution
• based on a set of recursive equations, derivation
  of
• - marginal queue length distribution pi(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)
• CmaxBi - ai, 1iM 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 p 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 , 1kK
• - decomposition-aggregation procedure
• p(S) Prob(S Ek) pa (Ek)
• - for symmetrical networks uniform
  conditional distribution
• Prob(S Ek) 1/ Ek
• - aggregated probabilitiies pa pa Qa
• - with aggregated matrix Qa qa (k,h)
• qa (k,h) (1/ Ek) Qkh 1T where 1
  (1,,1)
• computation of p reduces to the computation of
  pa O(K3 )

29
Example of symmetrical networks

• Each node has the same probabilistic behavior
• µi µ, Bi B, 1iM
• p1i ?0 ? p1m ((Im-1)mod M)1 ?0 , 1iM
  ,1mM-1
• pi j ?0 and pi k ?0 ? pi j pi k r , 1i,j,kM
• where r1/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
• - 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

31
Markov Process Decomposition

• Markov process with state space E and transition
  matrix Q
• Identify a partition of E into K subsets
• EU 1kK Ek
• ? decomposition of Q
• decomposition-aggregation procedure
• Prob(SEk) conditional distribution
• pa aggregated probabilities
• computation of p(S) reduces to
• the computation of Prob(S Ek) ? S, ?Ek
• the computation of pa
• exact computation soon becomes computationally
  intractable
• EXCEPT FOR special cases (symmetrical networks)
  
• approximation of Prob(S Ek) and Prob(Ek)

p(S) Prob(S Ek) pa (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 for closed QNB

• M exponential, G general, GE generalized
  exponential
• A/B/s Kendalls notation
• A customer interarrival time distribution
• B service time distribution
• s the number of identical servers

37
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 assumes generalized exponential
  service time and evaluates the queue length
  distribution and average performance indices

Methods
39
Algorithm for closed QNB comparison
Observations
40
Approximate methods for open QNB

• 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
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), 1iM
• T(i) corresponds to node i
• analysis of isolated subnetworks
• T(i) as a M/M/1/Bi queue TED
• T(i) as a M/PHn/1/Bi queue TPD
• efficient solution methods

Methods
42
Algorithm for open QNB comparison

• All algorithms evaluate for each node i
• pi queue length distribution
• Li mean queue length, Xi node throughput, Ri
  node mean response time

Observations
43
QNB equivalence Properties

• equivalencies in terms of
• - state probability distribution p
• - average performance indices
• passage time distribution

• most of the equivalencies derive from the
  identity of the network processes Þ p
• Remark even 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 p 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)

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
• µifi(ni) load dependent service rates, P
  routing matrix
• p state distribution
• exponential networks with RS-RD blocking
• fi(k) any positive arbitrary function for kgtBi
• hi ei yi , ei defined in formula F2
• y(y1,,yM), yy A, Aaij, aijpji , j?i,
  aii1-Sj?i aij 1i,jM

45
Equivalence between networks with different
blocking types
X and Y blocking types XY identity identical
p XY reducibility correspondence between
p 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
51
Example of heterogeneous QNB computer-communicat
ion system

• C1, C2 computer CPU subsystem RS-RD blocking
• D1, D2 computer Disk subsystem BAS blocking
• N1, N3 computer network access BAS blocking
• N2, 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 -gt product-form
  solution F2 - convolution algorithm

52
Conclusions and open research

• 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

53

• 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

54
References 1

• Books
• S.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 Issues
• S.Balsamo, D. Kouvatsos Special Issue "Queueing
  Networks with Blocking, Performance Evaluation
  Journal, 2003, 51/2-4.
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60
Additional method information

• Convolution algorithm for product-form closed QNB
• Details of the approximation algorithms for
• closed QNB
• open QNB
• Observations on method comparison
• Details on product-form conditions

61
Convolution algorithm for QNB

• for j1 to M
• for n a1 to B1
• for j2 to M
• MIN min (Aj-1Bj, B(j-1)aj)
• MAX max (Aj-1Bj, 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

Computation of values aj, Aj, B(j)
Initialisation
Computation of functions Gj(n)
Back
62
Cyclic Networks Throughput Approximation

• BBS or BAS blocking
• Assumption
• throughput 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 BAS
• BBS maximum throughput for NN
• N?????? B even
• N?????????????? B odd
• Exact computation of few values of X(N)
• Interpolation by
• X(N)X(N1)- y xN-N
• where yX(N) -X(B-)/? 1i(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 1iN Bi

1 N B
non-decreasing non-increasing
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63
Cyclic Networks Throughput Approximation

• BAS blocking
• conjectured maximum throughput for N N
  dependent on queue capacities and service rates
• N _at_ éS i(Bi1)/2 ù- 1
• More evaluation of X(N)
• Approximation for NNB-2
• Computational complexity gtgt exact analysis

1. Exact computation of X(N) for NB--1, B-, N.
for 1NB- network without blocking
product-form algorithm X(N) solution of the
associated Markov chain 2. Approximation of X(N)
for B-1NN-1. formulas and solution of the
fixed-point problem For BAS blocking two
additional steps 3. Exact computation of X(N)
for NB-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 N1NB-2,
as at step 2.
Back
64
Cyclic Networks - Network Decomposition

• BBS blocking
• 1.network decomposition M one-node subnetworks
• 2. analysis of isolated subnetworks
• M/M/1/Bi arrival rate li and load dependent
  service rate µi(n), 0nBi to derive the
  marginal queue length distribution pi (n),
  0nBi 1iM
• 3. approximate aggregation
• CASE B18
• µi(n)(1/µ i) S1IMb ij(n)Si1kj (1/µ
  k)-1
• µM(n)µM "n , " i
• l i X/(1-pi(Bi)) ()
• X network throughput
• b ij(n) probability that nodes i2,,j are full
  and node j1 is not full, given n customers in
  node i, 1i,jM
• b ij(n) in terms of pk (n), I1kM
• pi(Bi) function of l i
• Given X, l i is the fixed point of equation ()
• Iterative scheme

Back
65
Cyclic Networks - Network Decomposition

• Iterative scheme CASE B1 8
• CASE B1lt8
• Approximation of µM(n)µ and an additional
  iteration cycle to compute pi(Bi), " i
• Computational complexity O( k M4(B)3)
• operations for k iteration steps, Bmax iBi

Input approximate throughput Xmin(0),
Xmax(0) Repeat (step k1) computes new
parameters li and µi(n), 0nBi, 1iM
appropriately updates the k-th throughput
approximation Xmin(k), Xmax(k) Until
(Xmax(k)- Xmin(k))lte and S average nodes
population _at_ N and liltµi-1,"i Output
approximate throughput
Back
66
Cyclic Networks - Network Decomposition

• BBS blocking and assume B18
• network decomposition applied to nested
  subnetworks
• Ci has with load dependent service rate ni(n) and
  a variable queue capacity
• fi(nN) fraction of time in which the queue
  capacity is n, given N customers in the network
  1nN
• 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

Back
67
Cyclic Networks - Network Decomposition
Analysis of subnetwork M-1,M to define the
aggregate node CM-1, seen by node M-2 From
node iM-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

• Parameters ni(n) and fi(nN) of each VB
• two-node network are derived by the solution
• of 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 the
• IB model solution.
• Simple, non iterative algorithm
• Computational complexity O(MN3) operations

Back
68
Arbitrary Topology NetworksMatching State Space

• BAS blocking and exponential service time
• approximate the network with blocking with a
  network without blocking by choosing N to
  approximately match the state space cardinality
  of the underlying Markov chain
• Assumption
• the 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(M3MN2)

1. Compute K(N) by a convolution algorithm 2.
Determine N' to minimize K(N)-K'(N'), 1N'N,
by linear search in1,N 3. Compute the
throughput of the network without blocking by a
convolution algorithm
Back
69
Arbitrary Topology Networks- Approximate MVA

• BAS blocking and exponential service time
• Modified MVA algorithm defined for product-form
  networks with unlimited queue capacities and
  based on
• Little law
• arrival theorem which that does not hold for Q.N.
  with blocking
• Recursive scheme
• Ri(n) (1/µi) 1Li(n-1) 1iM (1)
• Xi(n) n ei/S 1jM ejRj (n) 1iM (2)
• Li(n) Xi (n) / Ri (n) 1iM (3)
• Simple implementation
• Computational complexity O(M3 k MN )

1. Initialization 2. For each population
n1,N Repeat computation of MVA equations
(1)-(3) where (1) is substituted by Ri(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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70
Arbitrary Topology Networks Maximum Entropy
Algorithm

• RS-RD blocking and generalized exponential
  service time
• Maximum entropy principle
• Approximation of p(S) for each network state
  S(n1,, nM) by maximizing the entropy function
• H(p)-SS p(S)log(p(S))
• subject to
• (I) (normalization) SS p(S)1
• (II) (probability of ni ai) Snigtaipi(ni)ui
• (III) (mean queue length) SainiBihi(ni)pi(ni)Li
  
• (IV) (full node) SainiBifi(ni)pi(ni)Fi
• aimax0, N-BBi minimum node i population
• hi(ni)min0, ni-ai-1 and f(ni)max0, ni-Bi1
• Product form approximation by the Lagrange's
  method of undetermined multipliers
• p(n)(1/Z) P1 i M xi(ni) yihi(ni ) zifi
• Z normalizing constant,xi(ni)1if niai,
  xi(ni)xi if ainiBi,
• xi, yi and zi are the Lagrangian coefficients
  corresponding to constraints (II)-(IV)

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71
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 expression
• Approximation of the closed network with a pseudo
  open without exogenous departures and arrivals
• Approximate analysis of the open network
• by adding the constraint NSi Li
• slight modifications to derive a solution for xi,
  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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72
Algorithm for closed QNB comparison

• Observations
• Cyclic 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 networks
• the time computational complexities of ND
  (O(kM4(B)3)) and of VQCD (O(MN3)) show a
  different dependence on network parameters. If
  NltMB 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 M20)
• ND and TA apply to a more general class than VQCD

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73
Algorithm for closed QNB comparison

• Observations
• Arbitrary Topology Networks
• compare the two approximation algorithms Matching
  State Space (MSS) and Approximate MVA (AMVA) that
  apply to the same networks
• MSS is more accurate than AMVA both in terms of
  average and maximum relative errors
• approximations are quite different, their
  rationales are not related
• both 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 AMVA
• both algorithms are stable
• The Maximum Entropy Algorithm applies to a more
  general class of networks

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74
Tandem Exponential Decomposition

• BAS blocking
• subsystem T(i) M/M/1/ Bi
• µu(1) l?, 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 2iM (1)
• µd(i) (1/µi)pb(i1) / µd(i1)
  -1 1iM-1 (2)
• X1X2XM
• 3 equivalent systems to determine the unknowns

0. Initialization µu(1)l, µd(i)µi "i 1.
Repeat 1a forward cycle for i1,,
M-1 compute ps(i) and µu(i1) by
(1) 1b backward cycle for iM,, 2 compute
pb(i) and µd(i-1) by (2) until maxXi-Xj,
1i,jMlte 2. Compute Li, pi (n), 0nBi, 1iM
and X1
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75
Tandem Phase-Type Decomposition

• BAS blocking
• subsystem T(i) M/ PHn /1/ Bi
• T(M) is M/M/1/BM 1 with service rate µM
• Service with M-i1 exponential phases
• to consider blocking due to
• nodes (i1,,M)
• PHn is represented by the pair (?i,Ti),
• ?i1,0,0,0 an (M-i1)-vector
• Tiµi, µI1,,µMTA, Aars (ir,sM) upper
  triangular square matrix of probabilities among
  the exponential phases
• aij?i1wi1(j) 1iM, i1jM (1)
• ?ipi(Bi1)/?iwiTi-1 1 2iM-1 (2)
• wi(j)pi(0)?iRiBi ?ij)/p i(Bi) 2iM-1,
  i1jM (3)
• wiwi(j) (i1jM) row (M-i1)-vector, 111T
• ?ij column (M-i1)-vectors with all 0 except 1 in
  j-th
• fixed point problem
• lili/1-p i(Bi1) (4)

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76
Tandem Phase-Type Decomposition

• If the first node has unlimited capacity (B1 )
• If the first node has finite capacity (B1lt ) ,
  the first arrival rate l1 ? l exogenous arrival
  rate
• Add an iterative cycle to estimate the effective
  arrival rates li for each node i
• Convergence has not been proved
• Computational complexity
• O(k1S2IM ki(M-i1)3Bi2)

• 1. Analysis of subsystem M
• 1a determine lM as the fixed point solution of
  (4)
• 1b compute pM by the M/M/1/BM1 analysis
• 1c compute jM-1µMpM(BM1)/lM
• 2. Analysis of subsystem T(i), iM-1,, 2
• 2a determine li as the fixed point solution of
  (4)
• 2b compute pi by the M/PHM-i1/1/Bi1 analysis
• 2c compute wi(j), ji and ai-1j, "j, by (3) (2)
  and (1)
• 3. Analysis of subsystem T(1)
• compute p1 by M/PHM/1/B11 analysis with arrival
  rate l

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77
Acyclic network decomposition

• BAS blocking, extension of the method of Tandem
  Exponential Decomposition
• Network decomposition into M one-node subsystems
  T(i) M/M/1/ Bi 1iM
• Uij pijgt0 predecessor of node I
• T(i) receives from Ui exponential sources with
  rates µuj(i), for jÎ Ui
• New set of equations to determine subnetwork
  parameters, new formulas for unknown rates µuj(i)
  and µd(i)
• pbj(ni) probability that at arrival time at
  T(i) from the j-th
• source n nodes are blocked by node i, 0nltUi,
  jÎ Ui
• ps(i) probability that at the end of a service
  T(i) is empty

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78
Acyclic network decomposition
0. Initialization µu1(1)li µd(i)µi I 1.
Repeat 1a forward cycle for i1,, M compute
ps(i) p(1i)/1-p(0i) and µuj(i) "j in U
j 1b backward cycle for iM,, 1 compute
pb(ni) and µd(i) until convergence of µd(i)
2. Compute Li, Xi , Ri and pi (n), 0nBi, 1iM

• Computational comp