On the subexponential properties in stationary single-server queues: a Palm-martingale approach

2004 ◽  
Vol 36 (03) ◽  
pp. 872-892
Author(s):  
Naoto Miyoshi
Keyword(s):  
Markov Chain ◽  
Marked Point ◽  
Marked Point Process ◽  
Single Server ◽  
Service Time Distribution ◽  
Single Server Queues ◽  
Stochastic Intensity ◽  
Finite State ◽  
Arrival Processes ◽  
Stationary Workload

This paper studies the subexponential properties of the stationary workload, actual waiting time and sojourn time distributions in work-conserving single-server queues when the equilibrium residual service time distribution is subexponential. This kind of problem has been previously investigated in various queueing and insurance risk settings. For example, it has been shown that, when the queue has a Markovian arrival stream (MAS) input governed by a finite-state Markov chain, it has such subexponential properties. However, though MASs can approximate any stationary marked point process, it is known that the corresponding subexponential results fail in the general stationary framework. In this paper, we consider the model with a general stationary input and show the subexponential properties under some additional assumptions. Our assumptions are so general that the MAS governed by a finite-state Markov chain inherently possesses them. The approach used here is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The proof is essentially an extension of the M/GI/1 case to cover ‘Poisson-like’ arrival processes such as Markovian ones, where the stochastic intensity is admitted.

On the subexponential properties in stationary single-server queues: a Palm-martingale approach

2004 ◽  
Vol 36 (3) ◽  
pp. 872-892 ◽  
Author(s):  
Naoto Miyoshi
Keyword(s):  
Markov Chain ◽  
Marked Point ◽  
Marked Point Process ◽  
Single Server ◽  
Service Time Distribution ◽  
Single Server Queues ◽  
Stochastic Intensity ◽  
Finite State ◽  
Arrival Processes ◽  
Stationary Workload

This paper studies the subexponential properties of the stationary workload, actual waiting time and sojourn time distributions in work-conserving single-server queues when the equilibrium residual service time distribution is subexponential. This kind of problem has been previously investigated in various queueing and insurance risk settings. For example, it has been shown that, when the queue has a Markovian arrival stream (MAS) input governed by a finite-state Markov chain, it has such subexponential properties. However, though MASs can approximate any stationary marked point process, it is known that the corresponding subexponential results fail in the general stationary framework. In this paper, we consider the model with a general stationary input and show the subexponential properties under some additional assumptions. Our assumptions are so general that the MAS governed by a finite-state Markov chain inherently possesses them. The approach used here is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The proof is essentially an extension of the M/GI/1 case to cover ‘Poisson-like’ arrival processes such as Markovian ones, where the stochastic intensity is admitted.

On the stationary workload distribution of work-conserving single-server queues: a general formula via stochastic intensity

2001 ◽  
Vol 38 (3) ◽  
pp. 793-798 ◽  
Author(s):  
Naoto Miyoshi
Keyword(s):  
Closed Form ◽  
Queueing System ◽  
Marked Point ◽  
Marked Point Process ◽  
Single Server ◽  
Stochastic Intensity ◽  
Preemptive Resume ◽  
Workload Distribution ◽  
Stationary Workload ◽  
Closed Form Formula

It is well known that a simple closed-form formula exists for the stationary distribution of the workload in M/GI/1 queues. In this paper, we extend this to the general stationary framework. Namely, we consider a work-conserving single-server queueing system, where the sequence of customers’ arrival epochs and their service times is described as a general stationary marked point process, and we derive a closed-form formula for the stationary workload distribution. The key to our proof is two-fold: one is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The other is the preemptive-resume last-come, first-served discipline.

On the stationary workload distribution of work-conserving single-server queues: a general formula via stochastic intensity

2001 ◽  
Vol 38 (03) ◽  
pp. 793-798
Author(s):  
Naoto Miyoshi
Keyword(s):  
Closed Form ◽  
Queueing System ◽  
Marked Point ◽  
Marked Point Process ◽  
Single Server ◽  
Stochastic Intensity ◽  
Preemptive Resume ◽  
Workload Distribution ◽  
Stationary Workload ◽  
Closed Form Formula

It is well known that a simple closed-form formula exists for the stationary distribution of the workload in M/GI/1 queues. In this paper, we extend this to the general stationary framework. Namely, we consider a work-conserving single-server queueing system, where the sequence of customers’ arrival epochs and their service times is described as a general stationary marked point process, and we derive a closed-form formula for the stationary workload distribution. The key to our proof is two-fold: one is the Palm-martingale calculus, that is, the connection between the notion of Palm probability and that of stochastic intensity. The other is the preemptive-resume last-come, first-served discipline.

A QUEUE WITH POSITIVE AND NEGATIVE ARRIVALS GOVERNED BY A MARKOV CHAIN

2003 ◽  
Vol 17 (4) ◽  
pp. 487-501 ◽  
Author(s):  
Yang Woo Shin ◽  
Bong Dae Choi
Keyword(s):  
Markov Chain ◽  
Transient State ◽  
Single Server ◽  
Negative Customers ◽  
Service Distribution ◽  
Finite State ◽  
Sojourn Time Distribution ◽  
Absorbing States ◽  
Negative Arrivals ◽  
Selection Of

We consider a single-server queue with exponential service time and two types of arrivals: positive and negative. Positive customers are regular ones who form a queue and a negative arrival has the effect of removing a positive customer in the system. In many applications, it might be more appropriate to assume the dependence between positive arrival and negative arrival. In order to reflect the dependence, we assume that the positive arrivals and negative arrivals are governed by a finite-state Markov chain with two absorbing states, say 0 and 0′. The epoch of absorption to the states 0 and 0′ corresponds to an arrival of positive and negative customers, respectively. The Markov chain is then instantly restarted in a transient state, where the selection of the new state is allowed to depend on the state from which absorption occurred.The Laplace–Stieltjes transforms (LSTs) of the sojourn time distribution of a customer, jointly with the probability that the customer completes his service without being removed, are derived under the combinations of service disciplines FCFS and LCFS and the removal strategies RCE and RCH. The service distribution of phase type is also considered.

Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

1997 ◽  
Vol 29 (04) ◽  
pp. 1039-1059
Author(s):  
Vinod Sharma
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Point Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Discrete Time Queues ◽  
Time Queues ◽  
Finite State

Recently, Asmussen and Koole (Journal of Applied Probability 30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

scholarly journals The Queueing Model MAP|PH|1|N with Feedback Operating in a Markovian Random Environment

2016 ◽  
Vol 34 (2) ◽  
Author(s):  
A.N. Dudin ◽  
A.V. Kazimirsky ◽  
V.I. Klimenok ◽  
L. Breuer ◽  
U. Krieger
Keyword(s):  
Random Environment ◽  
Queueing Systems ◽  
Arrival Process ◽  
Single Server ◽  
Service Time Distribution ◽  
Finite Buffer ◽  
Phase Type ◽  
Server Queue ◽  
Finite State ◽  
Markovian Random Environment

Queueing systems with feedback are well suited for the description of message transmission and manufacturing processes where a repeated service is required. In the present paper we investigate a rather general single server queue with a Markovian Arrival Process (MAP), Phase-type (PH) service-time distribution, a finite buffer and feedback which operates in a random environment. A finite state Markovian random environment affects the parameters of the input and service processes and the feedback probability. The stationary distribution of the queue and of the sojourn times as well as the loss probability are calculated. Moreover, Little’s law is derived.

Time-dependent rate conservation laws for a process defined with a stationary marked point process and their applications

1994 ◽  
Vol 31 (1) ◽  
pp. 114-129 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Conservation Laws ◽  
Point Process ◽  
Marked Point ◽  
Risk Process ◽  
Time Dependent ◽  
Marked Point Process ◽  
Single Server ◽  
Dependent Rate ◽  
Dependent Behavior ◽  
And Storage

We derive two kinds of rate conservation laws for describing the time-dependent behavior of a process defined with a stationary marked point process and starting at time 0. These formulas are called TRCLs (time-dependent rate conservation laws). It is shown that TRCLs are useful to study the transient behaviors of risk and storage processes with stationary claim and supply processes and with a general premium and release rates, respectively. Detailed discussions are given for the severity for the risk process, and for the workload process of a single-server queue.

Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

1997 ◽  
Vol 29 (4) ◽  
pp. 1039-1059
Author(s):  
Vinod Sharma
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Point Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Discrete Time Queues ◽  
Time Queues ◽  
Finite State

Recently, Asmussen and Koole (Journal of Applied Probability30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

scholarly journals Analysis of a Priority Queue with Phase-Type Service and Failures

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Alexander Dudin ◽  
Sergei Dudin
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Priority Queue ◽  
Arrival Process ◽  
Single Server ◽  
Service Time Distribution ◽  
New Class ◽  
Phase Type ◽  
Chain Problem ◽  
Insight Into

We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phase-type with failures (PHF) distribution, that generalizes the well-known phase-type (PH) distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of PHF type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service.

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