Queueing Systems

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

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_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