Single-station queueing system analysis.

The qsys module provides exact and approximate formulas for single-station queueing systems, including M/M/1, M/M/k, M/G/1, G/I/G/1 approximations, and finite buffer loss models.

Function List

Basic Queueing Models:

qsys_mm1 - M/M/1 queue: mean waiting time
qsys_mmk - M/M/k queue: mean waiting time with Erlang-C formula
qsys_mg1 - M/G/1 queue: Pollaczek-Khinchine formula
qsys_gm1 - G/M/1 queue: mean waiting time

Discrete-Time (Slotted) Models:

qsys_geogeo1 - Geo/Geo/1 queue: exact stationary distribution and means
qsys_geoxgeo1 - Geo^X/Geo/1 queue: batch arrivals, exact means via the pgf

G/I/G/1 Approximations:

qsys_gig1_ubnd_kingman - Kingman's upper bound for G/I/G/1
qsys_gig1_approx_allencunneen - Allen-Cunneen approximation
qsys_gig1_approx_gelenbe - Gelenbe approximation
qsys_gig1_approx_heyman - Heyman approximation
qsys_gig1_approx_kimura - Kimura approximation
qsys_gig1_approx_klb - Kramer-Langenbach-Belz approximation
qsys_gig1_approx_kobayashi - Kobayashi approximation
qsys_gig1_approx_marchal - Marchal approximation
qsys_gig1_approx_myskja - Myskja approximation
qsys_gig1_approx_myskja2 - Myskja improved approximation

G/I/G/k Approximations:

qsys_gigk_approx - G/I/G/k multiserver approximation
qsys_gigk_approx_kingman - Kingman approximation for G/I/G/k

Finite Buffer Loss Models:

qsys_mm1k_loss - M/M/1/K loss probability
qsys_mg1k_loss - M/G/1/K loss probability (Niu-Cooper)
qsys_mg1k_loss_mgs - M/G/1/K loss probability (MacGregor Smith)

qsys_mm1

Description: M/M/1 queue mean waiting time.

Syntax:

[W, rho] = qsys_mm1(lambda, mu)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate

Returns:

Name	Description
W	Mean waiting time (including service)
rho	Utilization

qsys_mmk

Description: M/M/k queue mean waiting time using Erlang-C formula.

Syntax:

[W, rho] = qsys_mmk(lambda, mu, k)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate per server
k	Number of servers

Returns:

Name	Description
W	Mean waiting time (including service)
rho	Utilization per server

qsys_mg1

Description: M/G/1 queue using the Pollaczek-Khinchine formula.

Syntax:

[W, rhohat] = qsys_mg1(lambda, mu, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
cs	Coefficient of variation of service time

Returns:

Name	Description
W	Mean waiting time (including service)
rhohat	Effective utilization for M/M/1 equivalence

qsys_geogeo1

Description: Discrete-time Geo/Geo/1 queue: exact stationary distribution and mean measures. Time advances in slots of unit length; in each slot an arrival occurs with probability a and, if the server is engaged, a service completion occurs with probability s. The observation epoch is selected by the convention argument: 'LAS_DA' (default, the state after both the departure and the arrival of the slot) or 'EAS' (the state after the departure and before the arrival). The two are one departure apart, so the mean sojourn times differ by exactly one slot and the waiting times coincide.

Syntax:

result = qsys_geogeo1(a, s)

result = qsys_geogeo1(a, s, convention)

Parameters:

Name	Description
a	Per-slot arrival probability, 0 < a < s
s	Per-slot service completion probability, 0 < s <= 1
convention	Observation epoch, 'LAS_DA' (default) or 'EAS'

Returns: struct with meanQueueLength, meanWaitingQueue, meanSojournTime, meanWaitingTime, meanServiceTime, utilization, throughput, emptyProb, ratio and a pmf function handle.

Reference: H. Daduna, Queueing Networks with Discrete Time Scale, LNCS 2046, Springer 2001, corollary 2.7.

qsys_geoxgeo1

Description: Discrete-time Geo^X/Geo/1 queue with batch arrivals. In each slot a batch arrives with probability a; the batch size is geometric on {1,2,...} with parameter beta, so the job arrival rate is lambda = a/beta and stability requires lambda < s. The mean follows from the queue-length generating function and depends on the batch only through its first two factorial moments. At beta = 1 the result equals qsys_geogeo1.

Syntax:

result = qsys_geoxgeo1(a, beta, s)

result = qsys_geoxgeo1(a, beta, s, convention)

Parameters:

Name	Description
a	Per-slot probability that a batch arrives, 0 < a <= 1
beta	Batch-size geometric parameter; batch supported on {1,2,...} with mean 1/beta
s	Per-slot service completion probability, 0 < s <= 1
convention	Observation epoch, 'LAS_DA' (default) or 'EAS'

Returns: struct with arrivalRate, throughput, utilization, boundaryEmptyProb, meanQueueLength, meanWaitingQueue, meanSojournTime, meanWaitingTime, meanServiceTime and a pgf function handle. No pmf is returned: for a general batch law the stationary distribution has no elementary closed form.

qsys_gm1

Description: G/M/1 queue mean waiting time.

Syntax:

W = qsys_gm1(sigma, mu)

Parameters:

Name	Description
sigma	Load at arrival instants (Laplace transform of inter-arrival times)
mu	Service rate

Returns:

Name	Description
W	Mean waiting time (including service)

qsys_gig1_ubnd_kingman

Description: Kingman's upper bound for G/I/G/1 queue.

Syntax:

[W, rhohat] = qsys_gig1_ubnd_kingman(lambda, mu, ca, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Upper bound on mean waiting time
rhohat	Effective utilization

qsys_gig1_approx_allencunneen

Description: Allen-Cunneen approximation for G/I/G/1 queue.

Syntax:

[W, rhohat] = qsys_gig1_approx_allencunneen(lambda, mu, ca, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Approximate mean waiting time
rhohat	Effective utilization

qsys_gig1_approx_gelenbe

Description: Gelenbe approximation for G/I/G/1 queue.

Syntax:

W = qsys_gig1_approx_gelenbe(lambda, mu, ca, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Approximate mean waiting time

qsys_gig1_approx_heyman

Description: Heyman approximation for G/I/G/1 queue.

Syntax:

[W, rhohat] = qsys_gig1_approx_heyman(lambda, mu, ca, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Approximate mean waiting time
rhohat	Effective utilization

qsys_gig1_approx_kimura

Description: Kimura approximation for G/I/G/1 queue.

Syntax:

W = qsys_gig1_approx_kimura(sigma, mu, ca, cs)

Parameters:

Name	Description
sigma	Load parameter
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Approximate mean waiting time

qsys_gig1_approx_klb

Description: Kramer-Langenbach-Belz (KLB) approximation for G/I/G/1 queue.

Syntax:

[W, rhohat] = qsys_gig1_approx_klb(lambda, mu, ca, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Approximate mean waiting time
rhohat	Effective utilization

qsys_gig1_approx_kobayashi

Description: Kobayashi approximation for G/I/G/1 queue.

Syntax:

[W, rhohat] = qsys_gig1_approx_kobayashi(lambda, mu, ca, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Approximate mean waiting time
rhohat	Effective utilization

qsys_gig1_approx_marchal

Description: Marchal approximation for G/I/G/1 queue.

Syntax:

[W, rhohat] = qsys_gig1_approx_marchal(lambda, mu, ca, cs)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times

Returns:

Name	Description
W	Approximate mean waiting time
rhohat	Effective utilization

qsys_gig1_approx_myskja

Description: Myskja approximation for G/I/G/1 queue.

Syntax:

W = qsys_gig1_approx_myskja(lambda, mu, ca, cs, q0, qa)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times
q0	Lowest value of the relative third moment
qa	Third relative moment E[X^3]/6/E[X]^3

Returns:

Name	Description
W	Approximate mean waiting time

qsys_gig1_approx_myskja2

Description: Myskja improved approximation for G/I/G/1 queue.

Syntax:

W = qsys_gig1_approx_myskja2(lambda, mu, ca, cs, q0, qa)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times
q0	Lowest value of the relative third moment
qa	Third relative moment E[X^3]/6/E[X]^3

Returns:

Name	Description
W	Approximate mean waiting time

qsys_gigk_approx

Description: G/I/G/k multiserver approximation.

Syntax:

[W, rhohat] = qsys_gigk_approx(lambda, mu, ca, cs, k)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate per server
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times
k	Number of servers

Returns:

Name	Description
W	Approximate mean waiting time
rhohat	Effective utilization

qsys_gigk_approx_kingman

Description: Kingman approximation for G/I/G/k multiserver queue.

Syntax:

[W, rhohat] = qsys_gigk_approx_kingman(lambda, mu, ca, cs, k)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate per server
ca	Coefficient of variation of inter-arrival times
cs	Coefficient of variation of service times
k	Number of servers

Returns:

Name	Description
W	Approximate mean waiting time
rhohat	Effective utilization

qsys_mm1k_loss

Description: M/M/1/K finite buffer loss probability.

Syntax:

[lossprob, rho] = qsys_mm1k_loss(lambda, mu, K)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
K	Buffer capacity (including server)

Returns:

Name	Description
lossprob	Blocking/loss probability
rho	Utilization

qsys_mg1k_loss

Description: M/G/1/K finite buffer loss probability using Niu-Cooper method.

Syntax:

[lossprob, rho] = qsys_mg1k_loss(lambda, svc_density, K)

Parameters:

Name	Description
lambda	Arrival rate
svc_density	Service time probability density function handle
K	Buffer capacity

Returns:

Name	Description
lossprob	Blocking/loss probability
rho	Utilization

qsys_mg1k_loss_mgs

Description: M/G/1/K finite buffer loss probability using MacGregor Smith approximation.

Syntax:

[lossprob, rho] = qsys_mg1k_loss_mgs(lambda, mu, mu_scv, K)

Parameters:

Name	Description
lambda	Arrival rate
mu	Service rate
mu_scv	Squared coefficient of variation of service time
K	Buffer capacity

Returns:

Name	Description
lossprob	Blocking/loss probability
rho	Utilization