171 research outputs found
A batch-service queueing model with a discrete batch Markovian arrival process
Queueing systems with batch service have been investigated extensively during the past decades. However, nearly all the studied models share the common feature that an uncorrelated arrival process is considered, which is unrealistic in several real-life situations. In this paper, we study a discrete-time queueing model, with a server that only initiates service when the amount of customers in system (system content) reaches or exceeds a threshold. Correlation is taken into account by assuming a discrete batch Markovian arrival process (D-BMAP), i.e. the distribution of the number of customer arrivals per slot depends on a background state which is determined by a first-order Markov chain. We deduce the probability generating function of the system content at random slot marks and we examine the influence of correlation in the arrival process on the behavior of the system. We show that correlation merely has a small impact on the threshold that minimizes the mean system content. In addition, we demonstrate that correlation might have a significant influence on the system content and therefore has to be included in the model
A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications
The main contribution of this paper is to present a new sufficient condition
for the subexponential asymptotics of the stationary distribution of a
GI/GI/1-type Markov chain without jumps from level "infinity" to level zero.
For simplicity, we call such Markov chains {\it GI/GI/1-type Markov chains
without disasters} because they are often used to analyze semi-Markovian queues
without "disasters", which are negative customers who remove all the customers
in the system (including themselves) on their arrivals. In this paper, we
demonstrate the application of our main result to the stationary queue length
distribution in the standard BMAP/GI/1 queue. Thus we obtain new asymptotic
formulas and prove the existing formulas under weaker conditions than those in
the literature. In addition, applying our main result to a single-server queue
with Markovian arrivals and the -bulk-service rule (i.e., MAP//1 queue), we obatin a subexponential asymptotic formula for the
stationary queue length distribution.Comment: Submitted for revie
Error bounds for last-column-block-augmented truncations of block-structured Markov chains
This paper discusses the error estimation of the last-column-block-augmented
northwest-corner truncation (LC-block-augmented truncation, for short) of
block-structured Markov chains (BSMCs) in continuous time. We first derive
upper bounds for the absolute difference between the time-averaged functionals
of a BSMC and its LC-block-augmented truncation, under the assumption that the
BSMC satisfies the general -modulated drift condition. We then establish
computable bounds for a special case where the BSMC is exponentially ergodic.
To derive such computable bounds for the general case, we propose a method that
reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to
level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the
properties of the bounds through the numerical results on an M/M/ retrial
queue, which is a representative example of LD-QBDs. Finally, we present
computable perturbation bounds for the stationary distribution vectors of
BSMCs.Comment: This version has fixed the bugs for the positions of Figures 1
through
Queues and risk processes with dependencies
We study the generalization of the G/G/1 queue obtained by relaxing the
assumption of independence between inter-arrival times and service
requirements. The analysis is carried out for the class of multivariate matrix
exponential distributions introduced in [12]. In this setting, we obtain the
steady state waiting time distribution and we show that the classical relation
between the steady state waiting time and the workload distributions re- mains
valid when the independence assumption is relaxed. We also prove duality
results with the ruin functions in an ordinary and a delayed ruin process.
These extend several known dualities between queueing and risk models in the
independent case. Finally we show that there exist stochastic order relations
between the waiting times under various instances of correlation
A discrete-time Markov modulated queuing system with batched arrivals
This paper examines a discrete-time queuing system with applications to
telecommunications traffic. The arrival process is a particular Markov
modulated process which belongs to the class of discrete batched Markovian
arrival processes. The server process is a single server deterministic queue. A
closed form exact solution is given for the expected queue length and delay. A
simple system of equations is given for the probability of the queue exceeding
a given length.Comment: to appear Performance Evaluatio
A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains
This paper proposes a new algorithm for computing the stationary distribution
vector in continuous-time upper block-Hessenberg Markov chains. To this end, we
consider the last-block-column-linearly-augmented (LBCL-augmented) truncation
of the (infinitesimal) generator of the upper block-Hessenberg Markov chain.
The LBCL-augmented truncation is a linearly-augmented truncation such that the
augmentation distribution has its probability mass only on the last block
column. We first derive an upper bound for the total variation distance between
the respective stationary distribution vectors of the original generator and
its LBCL-augmented truncation. Based on the upper bound, we then establish a
series of linear fractional programming (LFP) problems to obtain augmentation
distribution vectors such that the bound converges to zero. Using the optimal
solutions of the LFP problems, we construct a matrix-infinite-product (MIP)
form of the original (i.e., not approximate) stationary distribution vector and
develop a sequential update algorithm for computing the MIP form. Finally, we
demonstrate the applicability of our algorithm to BMAP/M/ queues and
M/M/ retrial queues.Comment: The typo in Abstract has been correcte
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