We consider a queuing model with the workload evolving between consecutive i.i.d. exponential timers $\{e_q^{(i)}\}_{i=1,2,...}$ according to a spectrally positive Lévy process $Y(t)$ which is reflected at 0. When the exponential clock $e_q^{(i)}$ ends, the additional state-dependent service requirement modifies the workload so that the latter is equal to $F_i(Y(e_q^{(i)}))$ at epoch $e^{(1)}_q+...+e^{(i)}_q$ for some random nonnegative i.i.d. functionals $F_i$. In particular, we focus on the case when $F_i(y)=(B_i-y)^+$, where $\{B_i\}_{i=1,2,...}$ are i.i.d. nonnegative random variables. We analyse the steady-state workload distribution for this model.
We study a many-server queuing system with general service time distribution and state dependent service rates. The dynamics of the system are modeled using measure valued processes which keep track of the residual service times. Under suitable conditions, we prove the existence of a unique fluid limit.
An infinite buffer batch service vacation queue has been studied where service rate of the batch is dependent on the size of the batch and vacation rate is dependent on the queue size at vacation initiation epoch. The arrivals follow the Markovian arrival process (MAP). For service rule, general bulk service (GBS) rule is considered. The service time and vacation time both are considered to be generally distributed. Several joint distributions of interest are obtained using the bivariate vector generating function method and the supplementary variable technique (SVT). Numerical results are pre…
We consider a model describing the waiting time of a server alternating between two service points. This model is described by a Lindley-type equation. We are interested in the time-dependent behaviour of this system and derive explicit expressions for its time-dependent waiting-time distribution, the correlation between waiting times, and the distribution of the cycle length. Since our model is closely related to Lindley's recursion, we compare our results to those derived for Lindley's recursion.
We discuss a single-server multi-station alternating queue where the preparation times and the service times are auto- and cross-correlated. We examine two cases. In the first case, preparation and service times depend on a common discrete time Markov chain. In the second case, we assume that the service times depend on the previous preparation time through their joint Laplace transform. The waiting time process is directly analysed by solving a Lindley-type equation via transform methods. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correla…
A single server retrial queueing system with two-classes of orbiting customers, and general class dependent service times is considered. If an arriving customer finds the server unavailable, it enters a virtual queue, called the orbit, according to its type. The customers from the orbits retry independently to access the server according to the constant retrial policy. We derive the generating function of the stationary distribution of the number of orbiting customers at service completion epochs in terms of the solution of a Riemann boundary value problem. For the symmetrical system we also d…
Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This paper analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service ($a,b$) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch u…
In this paper one presents method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically one considers inhomogeneous $M/M/S$ queueing system with possibly state-dependent arrival and service intensities and additionally possible batch arrivals and batch service. The unified approach based on logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side result, one shows by virtue of numerical…
Discrete-time queueing system has widespread applications in packet switching networks, internet protocol, Broadband Integrated Services Digital Network (B-ISDN), circuit switched time-division multiple access etc. In this paper, we analyze an infinite-buffer discrete-time group-arrival queue with single and multiple vacation policy where processing/transmission/service times dynamically vary with batch-size under service. The bivariate probability generating function of queue length and server content distribution has been derived first, and from that we extract the probabilities in terms of …
We study a generalization of the $M/G/1$ system (denoted by $rM/G/1$) with independent and identically distributed (iid) service times and with an arrival process whose arrival rate $λ_0f(r)$ depends on the remaining service time $r$ of the current customer being served. We derive a natural stability condition and provide a stationary analysis under it both at service completion times (of the queue length process) and in continuous time (of the queue length and the residual service time). In particular, we show that the stationary measure of queue length at service completion times is equal to…
This paper is devoted to the study of the number of customers in infinite-server systems driven by Hawkes processes. In these systems, the self-exciting arrival process is assumed to be represented by a Hawkes process and the self-exciting service process by a state-dependent Hawkes process (sdHawkes process). Under some suitable conditions, for the Hawkes/sdHawkes/infty system, the Markov property of the system is derived. The joint time-dependent distribution of the number of customers in the system, the arrival intensity and the server intensity is characterised by a system of differential …
Discrete-time queueing models find huge applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems, computer networks, etc. 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 …
Due to the widespread applicability of discrete-time queues in wireless networks or telecommunication systems, this paper analyzes an infinite-buffer batch-service queue with single and multiple vacation where customers/messages arrive according to the Bernoulii process and service time varies with the batch-size. The foremost focal point of this analysis is to get the complete joint distribution of queue length and server content at service completion epoch, for which first the bivariate probability generating function has been derived. We also acquire the joint distribution at arbitrary slot…
We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain the necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with t…
We establish stability criterion for a two-class retrial system with Poisson inputs, general class-dependent service times and class-dependent constant retrial rates. We also characterise an interesting phenomenon of partial stability when one orbit is tight but the other orbit goes to infinity in probability. All theoretical results are illustrated by numerical experiments.
We introduce a novel single-server queue with general retrial times and event-dependent arrivals. This is a versatile model for the study of service systems, in which the server needs a non-negligible time to retrieve waiting customers upon a service completion, while future arrivals depend on the last realized event. Such a model is motivated by the customers' behaviour in service systems where they decide to join based on the last realized event. We investigate the necessary and sufficient stability condition and derive the stationary distribution both at service completion epochs, and at an…
In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes c…
We study the sojourn time in a queueing system with a single exponential server, serving a Poisson stream of customers in order of arrival. Service is provided at low or high rate, which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, and otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace tran…
This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths…
We investigate the tail behaviour of the steady state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queuing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.
