A Convexity Property of a Markov-Modulated Queueing Loss System

1995 ◽  
Vol 9 (2) ◽  
pp. 193-199 ◽  
Author(s):  
Charles Du ◽  
Michael Pinedo
Keyword(s):  
Transition Probability ◽  
Loss Probability ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
Convexity Property ◽  
Loss System ◽  
Markov Modulated Poisson Process ◽  
Speed Parameter ◽  
Markov Modulated

In this note we consider a single-server queueing loss system with zero buffer. The arrival process is a nonstationary Markov-modulated Poisson process. The arrival process in state i is Poisson with rate λi. The process remains in state i for a time that is exponentially distributed with rate Cαi, with c being a control or speed parameter. The service rate in state i is exponentially distributed with rate μi. The process moves from state i to state j with transition probability qij. We are interested in the loss probability as a function of c. In this note we show that, under certain conditions, the loss probability decreases when the c increases. As such, this result generalizes a result obtained earlier by Fond and Ross.

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (4) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (04) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i . The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i . In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

A single-server queue with server vacations and a class of non-renewal arrival processes

1990 ◽  
Vol 22 (3) ◽  
pp. 676-705 ◽  
Author(s):  
David M. Lucantoni ◽  
Kathleen S. Meier-Hellstern ◽  
Marcel F. Neuts
Keyword(s):  
Waiting Times ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Arbitrary Time ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Server Queue ◽  
Arrival Processes ◽  
Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

scholarly journals Performance Measures of State Dependent MMPP/M/1 Queue

2018 ◽  
Vol 7 (4.10) ◽  
pp. 942 ◽  
Author(s):  
R. Sakthi ◽  
V. Vidhya ◽  
K. Mahaboob Hassain Sherieff ◽  
. .
Keyword(s):  
Performance Measures ◽  
Research Work ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
State Dependent ◽  
Markov Modulated ◽  
Service State ◽  
Modulated Process ◽  
Stationary Probability Vector

In this research work we are concerned with single unit server queue  queue with Markov Modulated process in Poisson fashion and the service time follow exponential distribution. The system is framed as a state dependent with the arrival process as Markov Modulated input and service is rendered by a single server with variation in service rate based on the intensity of service state of the system. The rate matrix that is essential to compute the stationary probability vector is obtained and various performance measures are computed using matrix method.

A monotonicity result for the workload in Markov-modulated queues

1998 ◽  
Vol 35 (03) ◽  
pp. 741-747 ◽  
Author(s):  
Nicole Bäuerle ◽  
Tomasz Rolski
Keyword(s):  
Poisson Process ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Convex Ordering ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated ◽  
Stationary Workload

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

A monotonicity result for the workload in Markov-modulated queues

1998 ◽  
Vol 35 (3) ◽  
pp. 741-747 ◽  
Author(s):  
Nicole Bäuerle ◽  
Tomasz Rolski
Keyword(s):  
Poisson Process ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Convex Ordering ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated ◽  
Stationary Workload

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

A single-server queue with server vacations and a class of non-renewal arrival processes

1990 ◽  
Vol 22 (03) ◽  
pp. 676-705 ◽  
Author(s):  
David M. Lucantoni ◽  
Kathleen S. Meier-Hellstern ◽  
Marcel F. Neuts
Keyword(s):  
Waiting Times ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Arbitrary Time ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Server Queue ◽  
Arrival Processes ◽  
Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes. Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

scholarly journals Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-$MAP/G_n^{(a,b)}/1$

2021 ◽  
Author(s):  
Umesh Chandra Gupta ◽  
Nitin Kumar ◽  
Sourav Pradhan ◽  
Farida Parvez Barbhuiya ◽  
Mohan L Chaudhry
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Service Time ◽  
Batch Size ◽  
Arrival Process ◽  
General Distribution ◽  
Complete Analysis ◽  
Service Rate ◽  
Single Server ◽  
Digital Platforms

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer's epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

Buffer overflow asymptotics for a buffer handling many traffic sources

1996 ◽  
Vol 33 (03) ◽  
pp. 886-903 ◽  
Author(s):  
Costas Courcoubetis ◽  
Richard Weber
Keyword(s):  
Generating Functions ◽  
Optimization Problem ◽  
Buffer Overflow ◽  
Buffer Size ◽  
Arrival Process ◽  
Service Rate ◽  
Atm Switch ◽  
Constant Rate ◽  
Source Models ◽  
Markov Modulated

As a model for an ATM switch we consider the overflow frequency of a queue that is served at a constant rate and in which the arrival process is the superposition of N traffic streams. We consider an asymptotic as N → ∞ in which the service rate Nc and buffer size Nb also increase linearly in N. In this regime, the frequency of buffer overflow is approximately exp(–NI(c, b)), where I(c, b) is given by the solution to an optimization problem posed in terms of time-dependent logarithmic moment generating functions. Experimental results for Gaussian and Markov modulated fluid source models show that this asymptotic provides a better estimate of the frequency of buffer overflow than ones based on large buffer asymptotics.

Work-modulated queues with applications to storage processes

1992 ◽  
Vol 29 (03) ◽  
pp. 699-712 ◽  
Author(s):  
Sid Browne ◽  
Karl Sigman
Keyword(s):  
Sufficient Conditions ◽  
Arrival Process ◽  
Service Rate ◽  
Single Server ◽  
Steady State Distribution ◽  
State Distribution ◽  
Deterministic Function ◽  
Customer Delay ◽  
Finite Moments ◽  
Storage Processes

We study two FIFO single-server queueing models in which both the arrival and service processes are modulated by the amount of work in the system. In the first model, the nth customer's service time, Sn , depends upon their delay, Dn , in a general Markovian way and the arrival process is a non-stationary Poisson process (NSPP) modulated by work, that is, with an intensity that is a general deterministic function g of work in system V(t). Some examples are provided. In our second model, the arrivals once again form a work-modulated NSPP, but, each customer brings a job consisting of an amount of work to be processed that is i.i.d. and the service rate is a general deterministic function r of work. This model can be viewed as a storage (dam) model (Brockwell et al. (1982)), but, unlike previous related literature, (where the input is assumed work-independent and stationary), we allow a work-modulated NSPP. Our approach involves an elementary use of Foster's criterion (via Tweedie (1976)) and in addition to obtaining new results, we obtain new and simplified proofs of stability for some known models. Using further criteria of Tweedie, we establish sufficient conditions for the steady-state distribution of customer delay and sojourn time to have finite moments.

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