queueing

queueing

1.2.8 2024-05-13

The queueing package provides functions for queueing networks and Markov chains analysis. This package can be used to compute steady-state performance measures for open, closed and mixed networks with single or multiple job classes. Mean Value Analysis (MVA), convolution, and various bounding techniques are implemented. Furthermore, several transient and steady-state performance measures for Markov chains can be computed, such as state occupancy probabilities, mean time to absorption, time-averaged sojourn times and so forth. Discrete- and continuous-time Markov chains are supported.

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Discrete-time Markov chains

dtmcchkP Check whether P is a valid transition probability matrix.
dtmc Compute stationary or transient state occupancy probabilities for a discrete-time Markov chain.
dtmcbd Returns the transition probability matrix P for a discrete birth-death process over state space {1, ..., N}.
dtmcexps Compute the expected number of visits to each state during the first N transitions, or until abrosption.
dtmcisir Check if P is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with transition matrix P.
dtmctaexps Compute the _time-averaged sojourn times_ 'M(i)', defined as the fraction of time spent in state i during the first n transitions (or until absorption), assuming that the state occupancy probabilities at time 0 are P0.
dtmcmtta Compute the expected number of steps before absorption for a DTMC with state space {1, ..., N} and transition probability matrix P.
dtmcfpt Compute mean first passage times and mean recurrence times for an irreducible discrete-time Markov chain over the state space {1, ..., N}.

Continuous-time Markov chains

ctmcchkQ If Q is a valid infinitesimal generator matrix, return the size (number of rows or columns) of Q.
ctmc Compute stationary or transient state occupancy probabilities for a continuous-time Markov chain.
ctmcbd Returns the infinitesimal generator matrix Q for a continuous birth-death process over the finite state space {1, ..., N}.
ctmcexps With three arguments, compute the expected times 'L(i)' spent in each state i during the time interval [0,t], assuming that the initial occupancy vector is P.
ctmcisir Check if Q is irreducible, and identify Strongly Connected Components (SCC) in the transition graph of the DTMC with infinitesimal generator matrix Q.
ctmctaexps Compute the _time-averaged sojourn time_ 'M(i)', defined as the fraction of the time interval [0,t] (or until absorption) spent in state i, assuming that the state occupancy probabilities at time 0 are P.
ctmcmtta Compute the Mean-Time to Absorption (MTTA) of the CTMC described by the infinitesimal generator matrix Q, starting from initial occupancy probabilities P.
ctmcfpt Compute mean first passage times for an irreducible continuous-time Markov chain.

Single Station Queueing Systems

qsmm1 Compute utilization, response time, average number of requests and throughput for a M/M/1 queue.
qsmmm Compute utilization, response time, average number of requests in service and throughput for a M/M/m queue, a queueing system with m identical servers connected to a single FCFS queue.
erlangb Compute the steady-state blocking probability in the Erlang loss model.
erlangc Compute the steady-state probability of delay in the Erlang delay model.
engset Evaluate the Engset loss formula.
qsmminf Compute utilization, response time, average number of requests and throughput for an infinite-server queue.
qsmm1k Compute utilization, response time, average number of requests and throughput for a M/M/1/K finite capacity system.
qsmmmk Compute utilization, response time, average number of requests and throughput for a M/M/m/K finite capacity system.
qsammm Compute _approximate_ utilization, response time, average number of requests in service and throughput for an asymmetric M/M/m queue.
qsmg1 Compute utilization, response time, average number of requests and throughput for a M/G/1 system.
qsmh1 Compute utilization, response time, average number of requests and throughput for a M/H_m/1 system.

Queueing Networks

qnclosed This function computes steady-state performance measures of closed queueing networks using the Mean Value Analysis (MVA) algorithm.
qncmmvabs Approximate Mean Value Analysis (MVA) for closed, multiclass queueing networks with K service centers and C customer classes.
qncmmva Compute steady-state performance measures for closed, multiclass queueing networks using the Mean Value Analysys (MVA) algorithm.
qncmmvaap This function is deprecated.
qncmnpop Given a network with C customer classes, this function computes the number of k-mixes 'H(r,k)' that can be constructed by the multiclass MVA algorithm by allocating k customers to the first r classes.
qncmpopmix Return the set of population mixes for a closed multiclass queueing network with exactly K customers.
qncmvisits Compute the average number of visits for the nodes of a closed multiclass network with K service centers and C customer classes.
qncscmva Conditional MVA (CMVA) algorithm, a numerically stable variant of MVA.
qncsconvld Convolution algorithm for product-form, single-class closed queueing networks with K general load-dependent service centers.
qncsconv Analyze product-form, single class closed networks with K service centers using the convolution algorithm.
qncsmvaap Analyze closed, single class queueing networks using the Approximate Mean Value Analysis (MVA) algorithm.
qncsmvablo Approximate MVA algorithm for closed queueing networks with blocking.
qncsmvald Mean Value Analysis algorithm for closed, single class queueing networks with K service centers and load-dependent service times.
qncsmva Analyze closed, single class queueing networks using the exact Mean Value Analysis (MVA) algorithm.
qncsvisits Compute the mean number of visits to the service centers of a single class, closed network with K service centers.
qnmarkov Compute utilization, response time, average queue length and throughput for open or closed queueing networks with finite capacity and a single class of requests.
qnmix Mean Value Analysis for mixed queueing networks.
qnmknode Creates a node; this function can be used together with 'qnsolve'.
qnom Exact analysis of open, multiple-class BCMP networks.
qnomvisits Compute the visit ratios to the service centers of an open multiclass network with K service centers and C customer classes.
qnopen Compute utilization, response time, average number of requests in the system, and throughput for open queueing networks.
qnos Analyze open, single class BCMP queueing networks with K service centers.
qnosvisits Compute the average number of visits to the service centers of a single class open Queueing Network with K service centers.
qnsolve High-level function for analyzing QN models.

Bounds Analysis

qnosaba Compute Asymptotic Bounds for open, single-class networks with K service centers.
qncsaba Compute Asymptotic Bounds for the system throughput and response time of closed, single-class networks with K service centers.
qnomaba Compute Asymptotic Bounds for open, multiclass networks with K service centers and C customer classes.
qncmaba Compute Asymptotic Bounds for closed, multiclass networks with K service centers and C customer classes.
qnosbsb Compute Balanced System Bounds for single-class, open networks with K service centers.
qncsbsb Compute Balanced System Bounds on system throughput and response time for closed, single-class networks with K service centers.
qncmbsb Compute Balanced System Bounds for closed, multiclass networks with K service centers and C customer classes.
qncmcb Compute Composite Bounds (CB) on system throughput and response time for closed multiclass networks.
qncspb Compute PB Bounds (C.
qncsgb Compute Geometric Bounds (GB) on system throughput, system response time and server queue lenghts for closed, single-class networks with K service centers and N requests.