api.dqsys

dqsys_geoxgeo1(a, beta, s, convention)

QSYS_GEOXGEO1 Exact analysis of a discrete-time Geo^X/Geo/1 batch queue.

RESULT = QSYS_GEOXGEO1(A, BETA, S) analyzes a slotted single-server queue with batch arrivals:

A - per-slot probability that a batch arrives (0 < A <= 1) BETA - batch-size geometric parameter; the batch is supported on

{1,2,…} with mean 1/BETA (0 < BETA <= 1)

S - per-slot service completion probability (0 < S <= 1)

Stability requires LAMBDA = A/BETA < S.

RESULT = QSYS_GEOXGEO1(A, BETA, S, CONVENTION) selects the observation epoch, ‘LAS_DA’ (default) or ‘EAS’; see QSYS_GEOGEO1 for what they mean.

With A(z) = 1-A+A*X(z) the pgf of the number of jobs arriving in one slot, the slot-boundary content obeys X(t+1) = X(t) - D(t) + A(t) with the departure resolved first, giving

P(z) = p_0 S (z-1) A(z) / ( z - A(z)(S+(1-S)z) ), p_0 = 1 - LAMBDA/S

and, on differentiating at z = 1,: E[N] = LAMBDA + ( A E[X(X-1)]/2 + LAMBDA(1-S) ) / (S - LAMBDA)

The batch enters only through its first two factorial moments. For the geometric batch, E[X] = 1/BETA and E[X(X-1)] = 2(1-BETA)/BETA^2. At BETA = 1 the batch is always a single job and the result equals QSYS_GEOGEO1.

Returns a struct with fields:

convention - Observation epoch used, ‘LAS_DA’ or ‘EAS’ batchArrivalProb - Per-slot probability that a batch arrives, A batchMean - Mean batch size E[X] batchSecondFactorialMoment - E[X(X-1)] serviceProb - Per-slot service completion probability S arrivalRate - Jobs arriving per slot, LAMBDA = A E[X] throughput - Departures per slot, LAMBDA utilization - Fraction of slots with the server serving, LAMBDA/S boundaryEmptyProb - Empty probability AT THE SLOT BOUNDARY, 1 - LAMBDA/S meanQueueLength - Mean number of jobs in the system meanWaitingQueue - Mean number of jobs waiting, i.e. not in service meanSojournTime - Mean sojourn time in slots, per job (not per batch) meanWaitingTime - Mean waiting time in slots, per job meanServiceTime - Mean service time, 1/S under LAS_DA, (1-S)/S under EAS pgf - Function handle P(z, Az) taking the slot-arrival pgf

value A(z) at the same z; defined for 0 < z <= 1

analyzer - Identifier string

No pmf is returned: for a general batch law the stationary distribution has no elementary closed form, so only the generating function is exact. Likewise BOUNDARYEMPTYPROB is deliberately epoch independent; the empty probability at the EAS epoch is p_0 + S p_1 and has no elementary form.

Examples

r = dqsys_geoxgeo1(0.1, 0.5, 0.9); % lambda = 0.2 r.arrivalRate % 0.2 r.meanQueueLength % 0.5143

Reference: H. Daduna, Queueing Networks with Discrete Time Scale, LNCS 2046, Springer 2001, chapter 6 for the discrete-time batch framework; the single-node pgf above is the standard discrete-time M/G/1-type derivation.

See also QSYS_GEOGEO1, QSYS_MXM1

dqsys_geogeo1(a, s, convention)

QSYS_GEOGEO1 Exact closed-form analysis of a discrete-time Geo/Geo/1 queue.

RESULT = QSYS_GEOGEO1(A, S) analyzes a slotted single-server queue with:: A - per-slot arrival probability (0 < A < S) S - per-slot service completion probability (0 < S <= 1)

RESULT = QSYS_GEOGEO1(A, S, CONVENTION) selects the observation epoch, ‘LAS_DA’ (default) or ‘EAS’. 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, independently of everything else. The system content at slot boundaries is a discrete birth-death chain whose stationary distribution is geometric.

The two conventions are not two systems. They are one system, Daduna’s LA-rule (events at the end of their slot) with the D/A-rule (departure resolved before arrival), observed at two instants. With X(t+1) = X(t) - D(t) + A(t), ‘LAS_DA’ is the law of X, taken after both events, and ‘EAS’ is the law of Y(t) = X(t) - D(t), taken after the departure and before the arrival. They are one departure apart, so the mean contents differ by exactly A and, by Little’s law, the sojourn times by one slot. The queueing delay is the same under both.

With rho = A/S and r = A(1-S)/(S(1-A)):

LAS_DA: p_0 = 1-rho, p_n = (1-rho) rho/(1-A) r^(n-1), n >= 1: E[N] = A(1-A)/(S-A), E[T] = (1-A)/(S-A)
EAS: p_n = (1-r) r^n, n >= 0: E[N] = A(1-S)/(S-A), E[T] = (1-S)/(S-A)

Both give U = rho, X = A and E[W] = A(1-S)/(S(S-A)).

Returns a struct with fields:: convention - Observation epoch used, ‘LAS_DA’ or ‘EAS’ arrivalProb - Per-slot arrival probability A serviceProb - Per-slot service completion probability S utilization - Fraction of slots with the server serving, A/S throughput - Departures per slot, A emptyProb - Probability the system is empty at the epoch ratio - Geometric decay ratio of the queue-length tail, r meanQueueLength - Mean number of jobs in the system meanWaitingQueue - Mean number of jobs waiting, i.e. not in service meanSojournTime - Mean sojourn time in slots meanWaitingTime - Mean waiting time in slots (epoch independent) meanServiceTime - Mean service time, 1/S under LAS_DA, (1-S)/S under EAS pmf - Function handle p(n) for the stationary queue length analyzer - Identifier string

Examples

r = dqsys_geogeo1(0.2, 0.5); r.meanQueueLength % 0.5333 r.meanSojournTime % 2.6667 r.pmf(0) % 0.6

Reference: H. Daduna, Queueing Networks with Discrete Time Scale, LNCS 2046, Springer 2001, corollary 2.7 (the LAS_DA branch reproduces it term for term with b = A, p = S, c = 1-A, q = 1-S).