BMAP/G/1 Queue Under D-Policy: Queue Length Analysis

ABSTRACT

We study the queue length distribution of a queueing system with BMAP arrivals under D-policy. The idle server begins to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We derive the vector generating functions of the queue lengths both at a departure and at an arbitrary point of time. Mean queue lengths are derived and a numerical example is presented.     

Queue Length Analysis of MAP/G/1 Queue Under D-Policy

Abstract

We study the queue length distribution of a queueing system with MAP arrivals under D-policy. The idle server begins to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We derive the vector generating functions of the queue lengths both at a departure and at an arbitrary point of time. Mean queue lengths will be derived from these transform results. A numerical example is provided.     

Workload Process,Waiting Times,and Sojourn Times in a Discrete Time MMAP[K]/SM[K]/1/FCFS Queue

Abstract

In this paper, we study the total workload process and waiting times in a queueing system with multiple types of customers and a first-come-first-served service discipline. An M/G/1 type Markov chain, which is closely related to the total workload in the queueing system, is constructed. A method is developed for computing the steady state distribution of that Markov chain. Using that steady state distribution, the distributions of total workload, batch waiting times, and waiting times of individual types of customers are obtained. Compared to the GI/M/1 and QBD approaches for waiting times and sojourn times in discrete time queues, the dimension of the matrix blocks involved in the M/G/1 approach can be significantly smaller.     

Transient analysis of a finite source discrete-time queueing system using homogeneous Markov system with state size capacities (HMS/c)

Abstract

In this article, a finite source discrete-time queueing system is modeled as a discrete-time homogeneous Markov system with finite state size capacities (HMS/c) and transition priorities. This Markov system is comprised of three states. The first state of the HMS/c corresponds to the source and the second one to the state with the servers. The second state has a finite capacity which corresponds to the number of servers. The members of the system which can not enter the second state, due to its finite capacity, enter the third state which represents the system's queue. In order to examine the variability of the state sizes recursive formulae for their factorial and mixed factorial moments are derived in matrix form. As a consequence the probability mass function of each state size can be evaluated. Also the expected time in queue is computed by means of the interval transition probabilities. The theoretical results are illustrated by a numerical example.     

Estimation of Measures in M/M/1 Queue

Maximum likelihood and uniform minimum variance unbiased estimators of steady-state probability distribution of system size, probability of at least ? customers in the system in steady state, and certain steady-state measures of effectiveness in the M/M/1 queue are obtained/derived based on observations on X, the number of customer arrivals during a service time. The estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures.     

Matrix–geometric analysis of the discrete time Gi/G/1 system

In this paper, we show that the discrete GI/G/1 system can be analysed as a QBD process with infinite blocks. Most importantly, we show that Matrix–geometric method can be used for analyzing this general queue system including establishing its stability criterion and for obtaining the explicit stationary probability and the waiting time distributions. This also settles the unwritten myth that Matrix–geometric method is limited to cases with at least one Markov based characterizing parameter, i.e. either interarrival or service times, in the case of queueing systems.     

AN INFINITE-PHASE QUASI-BIRTH-AND-DEATH MODEL FOR THE NON-PREEMPTIVE PRIORITY M/PH/1 QUEUE

This paper considers a single server queue that handles arrivals from N classes of customers on a non-preemptive priority basis. Each of the N classes of customers features arrivals from a Poisson process at rate λ _i and class-dependent phase type service. To analyze the queue length and waiting time processes of this queue, we derive a matrix geometric solution for the stationary distribution of the underlying Markov chain. A defining characteristic of the paper is the fact that the number of distinct states represented within the sub-level is countably infinite, rather than finite as is usually assumed. Among the results we obtain in the two-priority case are tractable algorithms for the computation of both the joint distribution for the number of customers present and the marginal distribution of low-priority customers, and an explicit solution for the marginal distribution of the number of high-priority customers. This explicit solution can be expressed completely in terms of the arrival rates and parameters of the two service time distributions. These results are followed by algorithms for the stationary waiting time distributions for high- and low-priority customers. We then address the case of an arbitrary number of priority classes, which we solve by relating it to an equivalent three-priority queue. Numerical examples are also presented.     

Estimation in M/Er /1 queueing systems

In this paper, the maximum likelihood estimates of the parameters for the M/E_r /1 queueing model are derived when the queue size at each departure point is observed. A numerical example is generated by simulating a finite Markov chain to illustrate the methodology for estimating the parameters with variable Erlang service time distribution. The problem of hypothesis testing and simultaneous Confidence regions of the parameter is also investigated.0     

ON THE MAP/G/1 QUEUE WITH LEBESGUE-DOMINATED SERVICE TIME DISTRIBUTION AND LCFS PREEMPTIVE REPEAT SERVICE DISCIPLINE

The present paper contains an analysis of the MAP/G/1 queue with last come first served (LCFS) preemptive repeat service discipline and Lebesgue-dominated service time distribution. The transient distribution is given in terms of a recursive formula. The stationary distribution as well as the stability condition are obtained by means of Markov renewal theory via a QBD representation of the embedded Markov chain at departures and arrivals.     

On Level Crossing Analysis of Queues

In this note we introduce a new level crossing analysis and using it derive an integral equation for the steady state waiting time in the GI/G/1 Queue. For the GI/M/1 queue we derive the rates of up- and down-crossings of the virtual delay process and two integral equations, one for the steady state time spent in the system and the other for the steady state waiting time in the queue. Also, the steady state probability distributions of the time spent in the system and the waiting time in the queue are obtained by solving these integral equations.     

A transient analysis of Markov fluid models with jumps

In this paper, we consider Markov fluid models with jumps which are useful for e.g. insurance risk modeling and the performance analysis of high-speed data networks. Recently, Ahn and Ramaswami [Ahn, S. & Ramaswami, V. (2004). Transient analysis of fluid flow models via stochastic coupling to a queue. Stochastic Models, 20 (1) 71–101] provided a transient analysis of the Markov modulated fluid flow model using stochastic coupling to a queueing model. Here we extend their results and provide a transient analysis of Markov fluid models with jumps. We also present some numerical examples.     

Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation

Abstract

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n(n > 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n ? 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.     

Markov-modulated infinite-server queues driven by a common background process

This paper studies a system with multiple infinite-server queues that are modulated by a common background process. If this background process, being modeled as a finite-state continuous-time Markov chain, is in state j, then the arrival rate into the i-th queue is λ_{i, j}, whereas the service times of customers present in this queue are exponentially distributed with mean μ^{? 1}_{i, j}; at each of the individual queues all customers present are served in parallel (thus reflecting their infinite-server nature).

Three types of results are presented: in the first place (i) we derive differential equations for the probability-generating functions corresponding to the distributions of the transient and stationary numbers of customers (jointly in all queues), then (ii) we set up recursions for the (joint) moments, and finally (iii) we establish a central limit theorem in the asymptotic regime in which the arrival rates as well as the transition rates of the background process are simultaneously growing large.     

The Waiting Time Distribution of a Type k Customer in a Discrete-Time MMAP[K]/PH[K]/c (c = 1, 2) Queue Using QBDs

Abstract

This paper presents an improved method to calculate the delay distribution of a type k customer in a first-come-first-serve (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements, and c servers, with c = 1, 2 (the MMAP[K]/PH[K]/c queue). The first algorithms to compute this delay distribution, using the GI/M/1 paradigm, were presented by Van Houdt and Blondia [Van Houdt, B.; Blondia, C. The delay distribution of a type k customer in a first come first served MMAP[K]/PH[K]/1 queue. J. Appl. Probab. 2002, 39 (1), 213–222; The waiting time distribution of a type k customer in a FCFS MMAP[K]/PH[K]/2 queue. Technical Report; 2002]. The two most limiting properties of these algorithms are: (i) the computation of the rate matrix R related to the GI/M/1 type Markov chain, (ii) the amount of memory needed to store the transition matrices A _l and B _l . In this paper we demonstrate that each of the three GI/M/1 type Markov chains used to develop the algorithms in the above articles can be reduced to a QBD with a block size which is only marginally larger than that of its corresponding GI/M/1 type Markov chain. As a result, the two major limiting factors of each of these algorithms are drastically reduced to computing the G matrix of the QBD and storing the 6 matrices that characterize the QBD. Moreover, these algorithms are easier to implement, especially for the system with c = 2 servers. We also include some numerical examples that further demonstrate the reduction in computational resources.     

Long-Range Dependence of Markov Renewal Processes

This paper examines long‐range dependence (LRD) and asymptotic properties of Markov renewal processes generalizing results of Daley for renewal processes. The Hurst index and discrepancy function, which is the difference between the expected number of arrivals in (0, t] given a point at 0 and the number of arrivals in (0, t] in the time stationary version, are examined in terms of the moment index. The moment index is the supremum of the set of r > 0 such that the rth moment of the first return time to a state is finite, employing the solidarity results of Sgibnev. The results are derived for irreducible, regular Markov renewal processes on countable state spaces. The paper also derives conditions to determine the moment index of the first return times in terms of the Markov renewal kernel distribution functions of the process.     

ANALYSIS OF A GI/G Y /1 SYSTEM IN DISCRETE-TIME

ABSTRACT

We consider a class of single server queueing systems in which customers arrive singly and service is provided in batches, depending on the number of customers waiting when the server becomes free. Service is independent of the batch size. This system could also be considered as a batch service queue in which a server visits the queue at arbitrary times and collects a batch of waiting customers for service, or waits for a customer to arrive if there are no waiting customers. A waiting server immediately collects and processes the first arriving customer. The system is considered in discrete time. The interarrival times of customers and the inter-visit times of the server, which we call the service time, have general distributions and are represented as remaining time Markov chains. We analyze this system using the matrix-geometric method and show that the resulting R matrix can be determined explicitly in some special cases and the stationary distributions are known semi-explicitly in some other special cases.     

Response Time Distribution in a D-MAP/PH/1 Queue with General Customer Impatience

ABSTRACT

This paper presents two methods to calculate the response time distribution of impatient customers in a discrete-time queue with Markovian arrivals and phase-type services, in which the customers’ patience is generally distributed (i.e., the D-MAP/PH/1 queue). The first approach uses a GI/M/1 type Markov chain and may be regarded as a generalization of the procedure presented in Van Houdt ^[14] Van Houdt , B. ; Lenin , R. B. ; Blondia , C. Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age dependent service times Queueing Systems and Applications 2003 , 45 1 , 59 – 73 . [CROSSREF]  [Google Scholar] for the D-MAP/PH/1 queue, where every customer has the same amount of patience. The key construction in order to obtain the response time distribution is to set up a Markov chain based on the age of the customer being served, together with the state of the D-MAP process immediately after the arrival of this customer. As a by-product, we can also easily obtain the queue length distribution from the steady state of this Markov chain.

We consider three different situations: (i) customers leave the system due to impatience regardless of whether they are being served or not, possibly wasting some service capacity, (ii) a customer is only allowed to enter the server if he is able to complete his service before reaching his critical age and (iii) customers become patient as soon as they are allowed to enter the server. In the second part of the paper, we reduce the GI/M/1 type Markov chain to a Quasi-Birth-Death (QBD) process. As a result, the time needed, in general, to calculate the response time distribution is reduced significantly, while only a relatively small amount of additional memory is needed in comparison with the GI/M/1 approach. We also include some numerical examples in which we apply the procedures being discussed.     

Stochastic monotonicity and comparability of Markov chains with block-monotone transition matrices and their applications to queueing systems

ABSTRACT

Motivated by various applications in queueing theory, this article is devoted to the stochastic monotonicity and comparability of Markov chains with block-monotone transition matrices. First, we introduce the notion of block-increasing convex order for probability vectors, and characterize the block-monotone matrices in the sense of the block-increasing order and block-increasing convex order. Second, we characterize the Markov chain with general transition matrix by martingale and provide a stochastic comparison of two block-monotone Markov chains under the two block-monotone orders. Third, the stochastic comparison results for the Markov chains corresponding to the discrete-time GI/G/1 queue with different service distributions under the two block-monotone orders are given, and the lower bound and upper bound of the Markov chain corresponding to the discrete-time GI/G/1 queue in the sense of the block-increasing convex order are found.     

Costing Mixed Coxian Phase-type Systems with Poisson Arrivals

Previously, we developed a modeling framework which classifies individuals with respect to their length of stay (LOS) in the transient states of a continuous-time Markov model with a single absorbing state; phase-type models are used for each class of the Markov model. We here add costs and obtain results for moments of total costs in (0, t], for an individual, a cohort arriving at time zero and when arrivals are Poisson. Based on stroke patient data from the Belfast City Hospital we use the overall modelling framework to obtain results for total cost in a given time interval.     

Delay analysis of a discrete-time GI − GI − 1 queue with reservation-based priority scheduling

In this paper, we study a discrete-time single-server queueing system where the arriving traffic consists of both delay-sensitive and delay-tolerant streams. We analyze the effects of a reservation-based scheduling discipline on the delay characteristics of both types of traffic. Under this discipline, a placeholder packet is inserted in the queue to accommodate future delay-sensitive arrivals, allowing them to bypass a part of the queue, without suppressing the delay-tolerant traffic.

On the basis of a probability generating functions (pgf) approach, we analyze the system state distribution and the delay distribution for either traffic type and present closed-form expressions for the expected delays as well as the tail probabilities of the packet delays. We illustrate the results comparing the delay performance of our reservation-based scheduling discipline to the performance achieved by the traditional Absolute Priority (AP) and First In First Out (FIFO) disciplines.