Product-Form Queueing Networks

MVA, convolution, and normalizing constant methods.

The pfqn module contains algorithms for product-form queueing networks, including Mean Value Analysis (MVA), convolution, and normalizing constant methods.

Function List

• pfqn_aql - Approximate Queue Length (AQL) algorithm for product-form networks.
• pfqn_bsfcfs - Bard-Schweitzer approximate MVA for FCFS scheduling with weighted priorities.
• pfqn_bs - Bard-Schweitzer Approximate Mean Value Analysis (MVA).
• pfqn_ca - Convolution Algorithm for exact normalizing constant computation.
• pfqn_cdfun - AMVA-QD class-dependence function for queue-dependent scaling.
• pfqn_comom - CoMoM algorithm for computing the normalizing constant.
• pfqn_comomrm_ld - CoMoM for repairman model with load-dependent service rates.
• pfqn_comomrm - CoMoM (Convolution Method of Moments) for finite repairman model.
• pfqn_comomrm_ms - CoMoM for multiserver repairman model.
• pfqn_comomrm_orig - Original CoMoM implementation for finite repairman model.
• pfqn_conwayms - Multiserver Linearizer approximation (Conway 1989).
• pfqn_cub - Cubature method for normalizing constant using Grundmann-Moeller rules.
• pfqn_egflinearizer - Extended generalized fixed-point Linearizer approximation.
• pfqn_fnc - Generate load-dependent rates for functional server model f(n)=n+c.
• pfqn_gflinearizer - Generalized fixed-point Linearizer with uniform scaling exponent.
• pfqn_gld - Exact normalizing constant for load-dependent queueing networks.
• pfqn_gldsingle - Exact normalizing constant for single-class load-dependent models.
• pfqn_grnmol - Normalizing constant using Grundmann-Moeller quadrature.
• pfqn_joint - Compute joint queue-length probability distribution.
• pfqn_kt - Knessl-Tier asymptotic expansion for normalizing constant.
• pfqn_lap - Laplace approximation for normalizing constant.
• pfqn_lcfsqn_ca - Convolution algorithm for 2-station LCFS queueing networks
• pfqn_lcfsqn_mva - Exact MVA for 2-station LCFS queueing networks.
• pfqn_le - Logistic expansion (LE) asymptotic approximation for normalizing constant.
• pfqn_linearizer - Linearizer approximation for single-server stations.
• pfqn_linearizerms - Multiserver Linearizer (Krzesinski/Conway/De Souza-Muntz).
• pfqn_linearizermx - Linearizer for mixed open/closed queueing networks.
• pfqn_lldfun - AMVA-QD load and queue-dependent scaling function.
• pfqn_ls - Logistic sampling approximation for normalizing constant.
• pfqn_mci - Monte Carlo Integration (MCI) for normalizing constant.
• pfqn_mmint2_gausslaguerre - McKenna-Mitra integral with Gauss-Laguerre quadrature.
• pfqn_mmint2_gausslegendre - McKenna-Mitra integral with Gauss-Legendre quadrature.
• pfqn_mmint2 - McKenna-Mitra integral form for repairman models using MATLAB integral.
• pfqn_mmsample2 - Monte Carlo sampling for repairman models using McKenna-Mitra form.
• pfqn_mu_ms - Compute load-dependent rates for m identical c-server FCFS stations.
• pfqn_mushift - Shift load-dependent service rate vector by removing first element.
• pfqn_mvald - Exact MVA for load-dependent closed queueing networks.
• pfqn_mvaldms - Load-dependent MVA for multiserver mixed networks (wrapper for pfqn_mvaldmx).
• pfqn_mvaldmx_ec - Compute effective capacity terms for MVALDMX solver.
• pfqn_mvaldmx - Load-dependent MVA for mixed open/closed networks with limited load dependence.
• pfqn_mva - Exact Mean Value Analysis (MVA) for product-form queueing networks.
• pfqn_mvams - General-purpose MVA for mixed networks with multiserver nodes.
• pfqn_mvamx - Exact MVA for mixed open/closed single-server networks.
• pfqn_lcfsqn_nc - Normalizing constant for LCFS queueing networks
• pfqn_ncld - Normalizing constant for load-dependent closed networks.
• pfqn_nc - Computes the normalizing constant of a product-form queueing network.
• pfqn_nc_sanitize - Sanitize and preprocess network parameters for NC solvers.
• pfqn_nrl - Normalizing constant via Normal Radius-Logistic (NRL) approximation.
• pfqn_nrp - Normalizing constant via Normal Radius-Probit (NRP) approximation.
• pfqn_panacea - PANACEA (PAth-based Normal Approximation for Closed networks Estimation Algorithm).
• pfqn_procomom2 - Product-form CoMoM for 2-station repairman model (queue + delay).
• pfqn_propfair - Proportionally fair allocation approximation (Walton 2009).
• pfqn_qd - Queue-Dependent (QD) approximate MVA solver.
• pfqn_qzgblow - Lower asymptotic bound on queue length (Zahorjan-Gittelsohn-Bryant).
• pfqn_qzgbup - Upper asymptotic bound on queue length (Zahorjan-Gittelsohn-Bryant).
• pfqn_rd - Reduced Decomposition (RD) method for load-dependent networks.
• pfqn_recal - RECAL (REcursive CALculation) method for normalizing constant.
• pfqn_schmidt - Schmidt's exact MVA for networks with general scheduling disciplines.
• pfqn_sqni - Single Queue Network Iteration (SQNI) approximate solver.
• pfqn_stdf_heur - Heuristic sojourn time distribution for multiserver FCFS nodes (McKenna 1987 variant).
• pfqn_stdf - Sojourn time distribution for multiserver FCFS nodes (McKenna 1987).
• pfqn_xzabalow - Lower asymptotic bound on throughput (Zahorjan-Balanced).
• pfqn_xzabaup - Upper asymptotic bound on throughput (Zahorjan-Balanced).
• pfqn_xzgsblow - Lower asymptotic bound on throughput (Zahorjan-Gittelsohn-Schweitzer-Bryant).
• pfqn_xzgsbup - Upper asymptotic bound on throughput (Zahorjan-Gittelsohn-Schweitzer-Bryant).

pfqn_aql

Description: Approximate Queue Length (AQL) algorithm for product-form networks.

Syntax:

[XN, QN, UN, RN, numIters, AN] = pfqn_aql(L, N, Z, TOL, MAXITER, QN0)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
TOL	Tolerance for convergence.
MAXITER	Maximum number of iterations.
QN0	Initial guess for queue lengths.

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
RN	Residence times.
numIters	Number of iterations.
AN	Average arrival rate at nodes.

pfqn_bsfcfs

Description: Bard-Schweitzer approximate MVA for FCFS scheduling with weighted priorities.

Syntax:

[XN, QN, UN, RN, it] = pfqn_bsfcfs(L, N, Z, tol, maxiter, QN, weight)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector (default: zeros).
tol	Convergence tolerance (default: 1e-6).
maxiter	Maximum number of iterations (default: 1000).
QN	Initial queue length matrix (default: uniform distribution).
weight	Weight matrix for relative priorities (default: ones).

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
RN	Residence times.
it	Number of iterations performed.

pfqn_bs

Description: Bard-Schweitzer Approximate Mean Value Analysis (MVA).

Syntax:

[XN, QN, UN, RN, it] = pfqn_bs(L, N, Z, tol, maxiter, QN0, type)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
tol	Tolerance for convergence.
maxiter	Maximum number of iterations.
QN0	Initial guess for queue lengths.
type	Scheduling strategy type (default: PS).

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
RN	Residence times.
it	Number of iterations performed.

pfqn_ca

Description: Convolution Algorithm for exact normalizing constant computation.

Syntax:

[Gn, lGn] = pfqn_ca(L, N, Z)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.

Returns:

Name	Description
Gn	Normalizing constant.
lGn	Logarithm of the normalizing constant.

pfqn_cdfun

Description: AMVA-QD class-dependence function for queue-dependent scaling.

Syntax:

pfqn_cdfun(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_comom

Description: CoMoM algorithm for computing the normalizing constant.

Syntax:

pfqn_comom(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_comomrm_ld

Description: CoMoM for repairman model with load-dependent service rates.

Syntax:

[G, lG, prob] = pfqn_comomrm_ld(L, N, Z, mu, options)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
mu	Load-dependent rate matrix (MxNt matrix).
options	Solver options.

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.
prob	State probability distribution.

pfqn_comomrm

Description: CoMoM (Convolution Method of Moments) for finite repairman model.

Syntax:

[lG, lGbasis] = pfqn_comomrm(L, N, Z, m, atol)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
m	Replication factor (default: 1).
atol	Absolute tolerance for numerical computations.

Returns:

Name	Description
lG	Logarithm of normalizing constant.
lGbasis	Logarithm of basis functions.

pfqn_comomrm_ms

Description: CoMoM for multiserver repairman model.

Syntax:

[G, lG, prob] = pfqn_comomrm_ms(L, N, Z, m, S)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
m	Replication factor (default: 1).
S	Number of servers at queueing stations.

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.
prob	State probability distribution.

pfqn_comomrm_orig

Description: Original CoMoM implementation for finite repairman model.

Syntax:

pfqn_comomrm_orig(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_conwayms

Description: Multiserver Linearizer approximation (Conway 1989).

Syntax:

[Q, U, R, C, X, totiter] = pfqn_conwayms(L, N, Z, nservers, type, tol, maxiter)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
nservers	Number of servers per station.
type	Scheduling strategy type per station (default: FCFS).
tol	Convergence tolerance (default: 1e-8).
maxiter	Maximum number of iterations (default: 1000).

Returns:

Name	Description
Q	Mean queue lengths.
U	Utilization.
R	Residence times.
C	Cycle times.
X	System throughput.
totiter	Total number of iterations.

pfqn_cub

Description: Cubature method for normalizing constant using Grundmann-Moeller rules.

Syntax:

[Gn, lGn] = pfqn_cub(L, N, Z, order, atol)

Parameters:

Name	Description
L	Service demand matrix (MxR).
N	Population vector (1xR).
Z	Think time vector (1xR).
order	Degree of cubature rule (default: ceil((sum(N)-1)/2)).
atol	Absolute tolerance (default: 1e-8).

Returns:

Name	Description
Gn	Estimated normalizing constant.
lGn	Logarithm of normalizing constant.

pfqn_egflinearizer

Description: Extended generalized fixed-point Linearizer approximation.

Syntax:

[Q, U, W, C, X, totiter] = pfqn_egflinearizer(L, N, Z, type, tol, maxiter, alpha)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
type	Scheduling strategy type per station.
tol	Convergence tolerance (default: 1e-8).
maxiter	Maximum number of iterations (default: 1000).
alpha	Per-class scaling exponent vector.

Returns:

Name	Description
Q	Mean queue lengths.
U	Utilization.
W	Waiting times.
C	Cycle times.
X	System throughput.
totiter	Total iterations performed.

pfqn_fnc

Description: Generate load-dependent rates for functional server model f(n)=n+c.

Syntax:

[mu, c] = pfqn_fnc(alpha, c)

Parameters:

Name	Description
alpha	Rate parameters (Mx N matrix).
c	Constant offset parameter (default: auto-determined).

Returns:

Name	Description
mu	Load-dependent service rates.
c	Determined offset constant.

pfqn_gflinearizer

Description: Generalized fixed-point Linearizer with uniform scaling exponent.

Syntax:

[Q, U, W, C, X, totiter] = pfqn_gflinearizer(L, N, Z, type, tol, maxiter, alpha)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
type	Scheduling strategy type per station.
tol	Convergence tolerance.
maxiter	Maximum number of iterations.
alpha	Uniform scaling exponent for all classes.

Returns:

Name	Description
Q	Mean queue lengths.
U	Utilization.
W	Waiting times.
C	Cycle times.
X	System throughput.
totiter	Total iterations performed.

pfqn_gld

Description: Exact normalizing constant for load-dependent queueing networks.

Syntax:

[G, lG] = pfqn_gld(L, N, mu, options)

Parameters:

Name	Description
L	Service demand matrix (MxR).
N	Population vector (1xR).
mu	Load-dependent rate matrix (Mx sum(N)).
options	Solver options.

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.

pfqn_gldsingle

Description: Exact normalizing constant for single-class load-dependent models.

Syntax:

[lG, G] = pfqn_gldsingle(L, N, mu, options)

Parameters:

Name	Description
L	Service demand vector (Mx1).
N	Population (scalar).
mu	Load-dependent rate matrix (MxN).
options	Solver options.

Returns:

Name	Description
lG	Logarithm of normalizing constant.
G	Normalizing constant.

pfqn_grnmol

Description: Normalizing constant using Grundmann-Moeller quadrature.

Syntax:

pfqn_grnmol(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_joint

Description: Compute joint queue-length probability distribution.

Syntax:

[F, A] = pfqn_joint(L, N, m)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
m	Cache capacity

Returns:

Name	Description
F	Flow or throughput values
A	Absorption matrix or aggregate matrix

pfqn_kt

Description: Knessl-Tier asymptotic expansion for normalizing constant.

Syntax:

[G, lG, X, Q] = pfqn_kt(L, N, Z)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector (default: zeros).

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.
X	System throughput.
Q	Mean queue lengths.

pfqn_lap

Description: Laplace approximation for normalizing constant.

Syntax:

logI = pfqn_lap(L, N, Z)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.

Returns:

Name	Description
logI	Logarithm of normalizing constant approximation.

pfqn_lcfsqn_ca

Description: Convolution algorithm for 2-station LCFS queueing networks

Syntax:

[G, V] = pfqn_lcfsqn_ca(alpha, beta, N)

Parameters:

Name	Description
alpha	Service rates at LCFS station (1xR vector).
beta	Service rates at LCFS-PR station (1xR vector).
N	Population vector (default: ones(1,R)).

Returns:

Name	Description
G	Normalizing constant.
V	Auxiliary normalization term.

pfqn_lcfsqn_mva

Description: Exact MVA for 2-station LCFS queueing networks.

Syntax:

[T, Q, U, B] = pfqn_lcfsqn_mva(alpha, beta, N)

Parameters:

Name	Description
alpha	Service rates at LCFS station (1xR vector).
beta	Service rates at LCFS-PR station (1xR vector).
N	Population vector (default: ones(1,R)).

Returns:

Name	Description
T	Throughput vector.
Q	Mean queue lengths (2xR matrix).
U	Utilization (2xR matrix).
B	Back probability matrix (2xR).

pfqn_le

Description: Logistic expansion (LE) asymptotic approximation for normalizing constant.

Syntax:

[Gn, lGn] = pfqn_le(L, N, Z)

Parameters:

Name	Description
L	Service demand matrix (MxR).
N	Population vector (1xR).
Z	Think time vector (1xR).

Returns:

Name	Description
Gn	Estimated normalizing constant.
lGn	Logarithm of normalizing constant.

pfqn_linearizer

Description: Linearizer approximation for single-server stations.

Syntax:

[Q, U, W, C, X, totiter] = pfqn_linearizer(L, N, Z, type, tol, maxiter)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
type	Scheduling strategy type per station.
tol	Convergence tolerance.
maxiter	Maximum number of iterations.

Returns:

Name	Description
Q	Mean queue lengths.
U	Utilization.
W	Waiting times.
C	Cycle times.
X	System throughput.
totiter	Total iterations performed.

pfqn_linearizerms

Description: Multiserver Linearizer (Krzesinski/Conway/De Souza-Muntz).

Syntax:

[Q, U, R, C, X, totiter] = pfqn_linearizerms(L, N, Z, nservers, type, tol, maxiter)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
nservers	Number of servers per station.
type	Scheduling strategy per station (default: PS).
tol	Convergence tolerance (default: 1e-8).
maxiter	Maximum number of iterations (default: 1000).

Returns:

Name	Description
Q	Mean queue lengths.
U	Utilization.
R	Residence times.
C	Cycle times.
X	System throughput.
totiter	Total iterations performed.

pfqn_linearizermx

Description: Linearizer for mixed open/closed queueing networks.

Syntax:

[QN, UN, WN, CN, XN, totiter] = pfqn_linearizermx(lambda, L, N, Z, nservers, type, tol, maxiter, method)

Parameters:

Name	Description
lambda	Arrival rate vector (inf for closed classes).
L	Service demand matrix.
N	Population vector (inf for open classes).
Z	Think time vector.
nservers	Number of servers per station.
type	Scheduling strategy per station.
tol	Convergence tolerance (default: 1e-8).
maxiter	Maximum iterations (default: 1000).
method	Linearizer variant ('lin', 'gflin', 'egflin', default: 'egflin').

Returns:

Name	Description
QN	Mean queue lengths.
UN	Utilization.
WN	Waiting times.
CN	Cycle times.
XN	System throughput.
totiter	Total iterations.

pfqn_lldfun

Description: AMVA-QD load and queue-dependent scaling function.

Syntax:

pfqn_lldfun(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_ls

Description: Logistic sampling approximation for normalizing constant.

Syntax:

[Gn, lGn] = pfqn_ls(L, N, Z, I)

Parameters:

Name	Description
L	Service demand matrix (MxR).
N	Population vector (1xR).
Z	Think time vector (1xR).
I	Number of samples (default: 1e5).

Returns:

Name	Description
Gn	Estimated normalizing constant.
lGn	Logarithm of normalizing constant.

pfqn_mci

Description: Monte Carlo Integration (MCI) for normalizing constant.

Syntax:

[G, lG, lZ] = pfqn_mci(D, N, Z, I, variant)

Parameters:

Name	Description
D	Service demand matrix.
N	Population vector.
Z	Think time vector.
I	Number of samples (default: 1e5).
variant	MCI variant ('mci', 'imci', 'rm', default: 'imci').

Returns:

Name	Description
G	Normalizing constant estimate.
lG	Logarithm of normalizing constant.
lZ	Individual random sample log values.

pfqn_mmint2_gausslaguerre

Description: McKenna-Mitra integral with Gauss-Laguerre quadrature.

Syntax:

[G, lG] = pfqn_mmint2_gausslaguerre(L, N, Z, m)

Parameters:

Name	Description
L	Service demand vector.
N	Population vector.
Z	Think time vector.
m	Replication factor (default: 1).

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.

pfqn_mmint2_gausslegendre

Description: McKenna-Mitra integral with Gauss-Legendre quadrature.

Syntax:

[G, lG] = pfqn_mmint2_gausslegendre(L, N, Z, m)

Parameters:

Name	Description
L	Service demand vector.
N	Population vector.
Z	Think time vector.
m	Replication factor (default: 1).

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.

pfqn_mmint2

Description: McKenna-Mitra integral form for repairman models using MATLAB integral.

Syntax:

[G, lG] = pfqn_mmint2(L, N, Z)

Parameters:

Name	Description
L	Service demand vector.
N	Population vector.
Z	Think time vector.

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.

pfqn_mmsample2

Description: Monte Carlo sampling for repairman models using McKenna-Mitra form.

Syntax:

[G, lG] = pfqn_mmsample2(L, N, Z, samples)

Parameters:

Name	Description
L	Service demand vector.
N	Population vector.
Z	Think time vector.
samples	Number of samples.

Returns:

Name	Description
G	Normalizing constant estimate.
lG	Logarithm of normalizing constant.

pfqn_mu_ms

Description: Compute load-dependent rates for m identical c-server FCFS stations.

Syntax:

pfqn_mu_ms(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_mushift

Description: Shift load-dependent service rate vector by removing first element.

Syntax:

pfqn_mushift(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_mvald

Description: Exact MVA for load-dependent closed queueing networks.

Syntax:

[XN, QN, UN, CN, lGN, isNumStable, pi] = pfqn_mvald(L, N, Z, mu, stabilize)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
mu	Load-dependent rate matrix (MxNt).
stabilize	Force non-negative probabilities (default: true).

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
CN	Cycle times.
lGN	Logarithm of normalizing constant evolution.
isNumStable	Numerical stability flag.
pi	Marginal queue-length probabilities.

pfqn_mvaldms

Description: Load-dependent MVA for multiserver mixed networks (wrapper for pfqn_mvaldmx).

Syntax:

[XN, QN, UN, CN, lGN] = pfqn_mvaldms(lambda, D, N, Z, S)

Parameters:

Name	Description
lambda	Arrival rate vector.
D	Service demand matrix.
N	Population vector.
Z	Think time vector.
S	Number of servers per station.

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization (adjusted for multiservers).
CN	Cycle times.
lGN	Logarithm of normalizing constant.

pfqn_mvaldmx_ec

Description: Compute effective capacity terms for MVALDMX solver.

Syntax:

[EC, E, Eprime, Lo] = pfqn_mvaldmx_ec(lambda, D, mu)

Parameters:

Name	Description
lambda	Arrival rate vector.
D	Service demand matrix.
mu	Load-dependent rate matrix.

Returns:

Name	Description
EC	Effective capacity matrix.
E	E-function values.
Eprime	E-prime function values.
Lo	Open class load vector.

pfqn_mvaldmx

Description: Load-dependent MVA for mixed open/closed networks with limited load dependence.

Syntax:

[XN, QN, UN, CN, lGN, Pc] = pfqn_mvaldmx(lambda, D, N, Z, mu, S)

Parameters:

Name	Description
lambda	Arrival rate vector.
D	Service demand matrix.
N	Population vector.
Z	Think time vector.
mu	Load-dependent rate matrix.
S	Number of servers per station.

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
CN	Cycle times.
lGN	Logarithm of normalizing constant.
Pc	Marginal queue-length probabilities.

pfqn_mva

Description: Exact Mean Value Analysis (MVA) for product-form queueing networks.

Syntax:

[XN, QN, UN, CN, lGN] = pfqn_mva(L, N, Z, mi)

Parameters:

Name	Description
L	Service demand matrix (M x R).
N	Population vector (1 x R).
Z	Think time vector (1 x R).
mi	(Optional) Server multiplicity vector (1 x M). Default: single servers.

Returns:

Name	Description
XN	System throughput (1 x R).
QN	Mean queue length (M x R).
UN	Utilization (M x R).
CN	Residence time (M x R).
lGN	Logarithm of the normalizing constant.

pfqn_mvams

Description: General-purpose MVA for mixed networks with multiserver nodes.

Syntax:

[XN, QN, UN, CN, lG] = pfqn_mvams(lambda, L, N, Z, mi, S)

Parameters:

Name	Description
lambda	Arrival rate vector.
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
mi	Queue replication factors (default: ones).
S	Number of servers per station (default: ones).

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
CN	Cycle times.
lG	Logarithm of normalizing constant.

pfqn_mvamx

Description: Exact MVA for mixed open/closed single-server networks.

Syntax:

[XN, QN, UN, CN, lGN] = pfqn_mvamx(lambda, D, N, Z, mi)

Parameters:

Name	Description
lambda	Arrival rate vector.
D	Service demand matrix.
N	Population vector.
Z	Think time vector.
mi	Queue replication factors (default: ones).

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
CN	Cycle times.
lGN	Logarithm of normalizing constant.

pfqn_lcfsqn_nc

Description: Normalizing constant for LCFS queueing networks

Syntax:

[G, Ax] = pfqn_lcfsqn_nc(alpha, beta, N)

Parameters:

Name	Description
alpha	Service rates at LCFS station (1xR vector).
beta	Service rates at LCFS-PR station (1xR vector).
N	Population vector (default: ones(1,R)).

Returns:

Name	Description
G	Normalizing constant.
Ax	Cell array of A matrices for each state.

pfqn_ncld

Description: Normalizing constant for load-dependent closed networks.

Syntax:

[lG, G, method] = pfqn_ncld(L, N, Z, mu, varargin)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
mu	Load-dependent rate matrix.
options	Optional solver parameters.

Returns:

Name	Description
lG	Logarithm of normalizing constant.
G	Normalizing constant.
method	Method used for computation.

pfqn_nc

Description: Computes the normalizing constant of a product-form queueing network.

Syntax:

[lG, X, Q, method] = pfqn_nc(lambda, L, N, Z, varargin)

Parameters:

Name	Description
lambda	Arrival rates for open classes.
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
options	Optional arguments for solver options (SolverNC.defaultOptions).

Returns:

Name	Description
lG	Logarithm of the normalizing constant.
X	System throughputs.
Q	Mean queue lengths.
method	The specific method used for computation (e.g., 'exact', 'ca', 'recal').

pfqn_nc_sanitize

Description: Sanitize and preprocess network parameters for NC solvers.

Syntax:

[lambda, L, N, Z, lGremaind] = pfqn_nc_sanitize(lambda, L, N, Z, atol)

Parameters:

Name	Description
lambda	Arrival rate vector.
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
atol	Absolute tolerance.

Returns:

Name	Description
lambda	Sanitized arrival rates.
L	Sanitized service demands (rescaled).
N	Sanitized populations.
Z	Sanitized think times (rescaled).
lGremaind	Log normalization factor from removed classes.

pfqn_nrl

Description: Normalizing constant via Normal Radius-Logistic (NRL) approximation.

Syntax:

lG = pfqn_nrl(L, N, Z, alpha, options)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
alpha	Load-dependent rate matrix.
options	Solver options.

Returns:

Name	Description
lG	Logarithm of normalizing constant.

pfqn_nrp

Description: Normalizing constant via Normal Radius-Probit (NRP) approximation.

Syntax:

lG = pfqn_nrp(L, N, Z, alpha, options)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
alpha	Load-dependent rate matrix.
options	Solver options.

Returns:

Name	Description
lG	Logarithm of normalizing constant.

pfqn_panacea

Description: PANACEA (PAth-based Normal Approximation for Closed networks Estimation Algorithm).

Syntax:

[Gn, lGn] = pfqn_panacea(L, N, Z)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.

Returns:

Name	Description
Gn	Normalizing constant.
lGn	Logarithm of normalizing constant.

pfqn_procomom2

Description: Product-form CoMoM for 2-station repairman model (queue + delay).

Syntax:

[pk, lG, G, T, F, B] = pfqn_procomom2(L, N, Z, mu, m)

Parameters:

Name	Description
L	Service demand vector.
N	Population vector.
Z	Think time vector.
mu	Load-dependent rates (optional).
m	Replication factor (default: 1).

Returns:

Name	Description
pk	Marginal state probabilities.
lG	Logarithm of normalizing constant.
G	Normalizing constant.
T	Transfer matrices.
F	Product transfer matrix.
B	Combined transfer matrix.

pfqn_propfair

Description: Proportionally fair allocation approximation (Walton 2009).

Syntax:

[G, lG, Xasy] = pfqn_propfair(L, N, Z)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.

Returns:

Name	Description
G	Normalizing constant estimate.
lG	Logarithm of normalizing constant.
Xasy	Asymptotic throughput vector.

pfqn_qd

Description: Queue-Dependent (QD) approximate MVA solver.

Syntax:

[Q, X, U, iter] = pfqn_qd(L, N, ga, be, Q0)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
ga	Gamma scaling function (default: ones).
be	Beta scaling function (default: ones).
Q0	Initial queue length estimate (optional).

Returns:

Name	Description
Q	Mean queue lengths.
X	System throughput.
U	Utilization.
iter	Number of iterations performed.

pfqn_qzgblow

Description: Lower asymptotic bound on queue length (Zahorjan-Gittelsohn-Bryant).

Syntax:

pfqn_qzgblow(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_qzgbup

Description: Upper asymptotic bound on queue length (Zahorjan-Gittelsohn-Bryant).

Syntax:

pfqn_qzgbup(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_rd

Description: Reduced Decomposition (RD) method for load-dependent networks.

Syntax:

[lGN, Cgamma] = pfqn_rd(L, N, Z, mu, options)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector.
mu	Load-dependent rate matrix.
options	Solver options.

Returns:

Name	Description
lGN	Logarithm of normalizing constant.
Cgamma	Gamma correction factor.

pfqn_recal

Description: RECAL (REcursive CALculation) method for normalizing constant.

Syntax:

[G, lG] = pfqn_recal(L, N, Z, m0)

Parameters:

Name	Description
L	Service demand matrix.
N	Population vector.
Z	Think time vector (default: zeros).
m0	Initial multiplicity vector (default: ones).

Returns:

Name	Description
G	Normalizing constant.
lG	Logarithm of normalizing constant.

pfqn_schmidt

Description: Schmidt's exact MVA for networks with general scheduling disciplines.

Syntax:

[XN, QN, UN, CN, T] = pfqn_schmidt(D, N, S, sched)

Parameters:

Name	Description
D	Service demand matrix.
N	Population vector.
S	Number of servers per station (matrix or vector).
sched	Scheduling discipline per station.

Returns:

Name	Description
XN	System throughput.
QN	Mean queue lengths.
UN	Utilization.
CN	Cycle times.
T	Results table.

pfqn_sqni

Description: Single Queue Network Iteration (SQNI) approximate solver.

Syntax:

[Q, U, X] = pfqn_sqni(N, L, Z)

Parameters:

Name	Description
N	Population vector.
L	Service demand vector.
Z	Think time vector.

Returns:

Name	Description
Q	Mean queue lengths.
U	Utilization.
X	System throughput.

pfqn_stdf_heur

Description: Heuristic sojourn time distribution for multiserver FCFS nodes (McKenna 1987 variant).

Syntax:

pfqn_stdf_heur(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_stdf

Description: Sojourn time distribution for multiserver FCFS nodes (McKenna 1987).

Syntax:

pfqn_stdf(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_xzabalow

Description: Lower asymptotic bound on throughput (Zahorjan-Balanced).

Syntax:

XN = pfqn_xzabalow(L, N, Z)

Parameters:

Name	Description
L	Service demand vector.
N	Population.
Z	Think time.

Returns:

Name	Description
XN	Lower bound on throughput.

pfqn_xzabaup

Description: Upper asymptotic bound on throughput (Zahorjan-Balanced).

Syntax:

XN = pfqn_xzabaup(L, N, Z)

Parameters:

Name	Description
L	Service demand vector.
N	Population.
Z	Think time.

Returns:

Name	Description
XN	Upper bound on throughput.

pfqn_xzgsblow

Description: Lower asymptotic bound on throughput (Zahorjan-Gittelsohn-Schweitzer-Bryant).

Syntax:

pfqn_xzgsblow(...)

Parameters:

Name	Description

Returns:

Name	Description

pfqn_xzgsbup

Description: Upper asymptotic bound on throughput (Zahorjan-Gittelsohn-Schweitzer-Bryant).

Syntax:

pfqn_xzgsbup(...)

Parameters:

Name	Description

Returns:

Name	Description