Matrix-Analytic Methods

QBD processes and matrix-analytic solutions.

The mam module implements matrix-analytic methods for analyzing queues with structured Markov chains, including QBD, MAP, RAP, and G/M/1 and M/G/1 type processes.

Function List

• aph_bernstein - Fits an APH distribution using Bernstein's approximation
• map_ccdf_derivative - Derivative at zero of a MAP CCDF
• map_jointpdf_derivative - Partial derivative at zero of a MAP's joint PDF
• mmap_compress - Compresses an MMAP into a smaller representation
• qbd_bmapbmap1 - Computes QBD blocks for a BMAP/BMAP/1 queue
• qbd_mapmap1 - Solves a MAP/MAP/1 queue using QBD methods
• qbd_raprap1 - Solves a RAP/RAP/1 queue using QBD methods
• qbd_rg - Computes QBD matrices R and G for MAP/MAP/1 queue
• qbd_R_logred - Computes the rate matrix R using logarithmic reduction
• qbd_R - Computes the rate matrix R using successive substitutions
• qbd_setupdelayoff - Queue-length analysis for system with setup delay and turn-off
• qbd_setupdelayoff_tmp - Alternative implementation for setup delay and turn-off analysis

aph_bernstein

Description: Fits an APH distribution using Bernstein's approximation

Syntax:

APH = aph_bernstein(f, order)

Parameters:

Name	Description
f	Fraction or frequency values
order	Order of approximation or moment

Returns:

Name	Description
APH	Acyclic phase-type distribution

map_ccdf_derivative

Description: Derivative at zero of a MAP CCDF

Syntax:

map_ccdf_derivative(...)

map_jointpdf_derivative

Description: Partial derivative at zero of a MAP's joint PDF

Syntax:

map_jointpdf_derivative(...)

mmap_compress

Description: Compresses an MMAP into a smaller representation

Syntax:

mmap_compress(...)

qbd_bmapbmap1

Description: Computes QBD blocks for a BMAP/BMAP/1 queue

Syntax:

function qbd_bmapbmap1(MAPa, pbatcha, MAPs)

Parameters:

Name	Description
MAPa	Markovian Arrival Process for arrivals
pbatcha	Batch arrival probability
MAPs	Markovian Arrival Process for service

qbd_mapmap1

Description: Solves a MAP/MAP/1 queue using QBD methods

Syntax:

[XN, QN, UN, pqueue, R, eta, G, A_1, A0, A1, U, MAPs] = qbd_mapmap1(MAPa, MAPs, util)

Parameters:

Name	Description
MAPa	Markovian Arrival Process for arrivals
MAPs	Markovian Arrival Process for service
util	Utilization values

Returns:

Name	Description
XN	System throughput
QN	Mean queue lengths
UN	Utilization
pqueue	Queue state probabilities
R	R matrix (QBD) or response times
eta	Adjustment factors
G	Normalizing constant
A_1	Transition rate matrix level -1
A0	Exit rate vector (phase-type)
A1	Transition rate matrix level 1
U	Utilization matrix
MAPs	Markovian Arrival Process representation

qbd_raprap1

Description: Solves a RAP/RAP/1 queue using QBD methods

Syntax:

[XN, QN, UN, pqueue, R, eta, G, B, L, F] = qbd_raprap1(RAPa, RAPs, util)

Parameters:

Name	Description
RAPa	Rational Arrival Process for arrivals
RAPs	Rational Arrival Process for service
util	Utilization values

Returns:

Name	Description
XN	System throughput
QN	Mean queue lengths
UN	Utilization
pqueue	Queue state probabilities
R	R matrix (QBD) or response times
eta	Adjustment factors
G	Normalizing constant
B	Boundary matrix or batch rate matrix
L	Service demand matrix or mean queue length
F	Flow or throughput values

qbd_rg

Description: Computes QBD matrices R and G for MAP/MAP/1 queue

Syntax:

[R, G, B, L, F, U] = qbd_rg(MAPa, MAPs, util)

Parameters:

Name	Description
MAPa	Markovian Arrival Process for arrivals
MAPs	Markovian Arrival Process for service
util	Utilization values

Returns:

Name	Description
R	R matrix (QBD) or response times
G	Normalizing constant
B	Boundary matrix or batch rate matrix
L	Service demand matrix or mean queue length
F	Flow or throughput values
U	Utilization matrix

qbd_R_logred

Description: Computes the rate matrix R using logarithmic reduction

Syntax:

qbd_R_logred(...)

qbd_R

Description: Computes the rate matrix R using successive substitutions

Syntax:

qbd_R(...)

qbd_setupdelayoff

Description: Queue-length analysis for system with setup delay and turn-off

Syntax:

qbd_setupdelayoff(...)

qbd_setupdelayoff_tmp

Description: Alternative implementation for setup delay and turn-off analysis

Syntax:

QN = qbd_setupdelayoff_tmp(lambda, mu, alpharate, alphascv, betarate, betascv)

Parameters:

Name	Description
lambda	Arrival rate(s)
mu	Load-dependent service rate matrix
alpharate	Alpha phase service rates
alphascv	Squared coefficient of variation for alpha phase
betarate	Beta phase service rates
betascv	Squared coefficient of variation for beta phase

Returns:

Name	Description
QN	Mean queue lengths