State-dependent Arrival Research Articles

Extended Conway-Maxwell-Poisson distribution and its properties and applications

A new three parameter natural extension of the Conway-Maxwell-Poisson (COM-Poisson) distribution is proposed. This distribution includes the recently proposed COM-Poisson type negative binomial (COM-NB) distribution [Chakraborty, S. and Ong, S. H. (2014): A COM-type Generalization of the Negative Binomial Distribution, Accepted in Communications in Statistics-Theory and Methods] and the generalized COM-Poisson (GCOMP) distribution [Imoto, T. :(2014) A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution, Applied Mathematics and Computation, 247, 824-834]. The proposed distribution is derived from a queuing system with state dependent arrival and service rates and also from an exponential combination of negative binomial and COM-Poisson distribution. Some distributional, reliability and stochastic ordering properties are investigated. Computational asymptotic approximations, different characterizations, parameter estimation and data fitting example also discussed.

State-dependent M/G/1 queueing systems

We consider a state-dependent $$M_{n}$$Mn/$$G_{n}$$Gn/1 queueing system with both finite and infinite buffer sizes. We allow the arrival rate of customers to depend on the number of people in the system. Service times are also state dependent and service rates can be modified at both arrivals and departures of customers. We show that the steady-state solution of this system at arbitrary times can be derived using the supplementary variable method, and that the system's state at arrival epochs can be analyzed using an embedded Markov chain. For the system with infinite buffer size, we first obtain an expression for the steady-state distribution of the number of customers in the system at both arbitrary and arrival times. Then, we derive the average service time of a customer observed at both arbitrary times and arrival epochs. We show that our state-dependent queueing system is equivalent to a Markovian birth-and-death process. This equivalency demonstrates our main insight that the $$M_{n}$$Mn/$$G_{n}$$Gn/1 system can be decomposed at any given state as a Markovian queue. Thus, many of the existing results for systems modeled as an M / M / 1 queue can be carried through to the much more practical M / G / 1 model with state-dependent arrival and service rates. Then, we extend the results to the $$M_{n}$$Mn/$$G_{n}$$Gn/1 queueing systems with finite buffer size.

Analysis of a retrial queue with multiple vacations and state dependent arrivals

This paper examines an M/G/1 retrial queueing system with multiple vacations and different arrival rates. Whenever the system is empty, the server immediately takes a vacation. At a vacation completion epoch, if the number of customers in the orbit is at least one the server remains in the system to activate service, otherwise the server avails multiple vacations until at least one customer is recorded in the orbit. The primary arrival rate is λ 1 when the server in idle and the primary arrival rate is λ 2 when the server is busy or on vacation (λ 1 > λ 2 ). The steady state queue size distribution of number of customers in the retrial group, expected number of customers in the retrial group and expected number of customers in the system are obtained. Some special cases are also discussed. Numerical illustrations are also provided.

Non-Markovian State-Dependent Networks in Critical Loading

□ We establish a heavy traffic limit theorem for the queue-length process in a critically loaded single class queueing network with state-dependent arrival and service rates. A distinguishing feature of our model is non-Markovian state dependence. The limit stochastic process is a continuous-path reflected process on the nonnegative orthant. We give an application to a generalized Jackson network with state-dependent rates.

Multi-channel Queues with Setup Time

Many practical queuing situations with congestion control mechanism due to high throughput demands in telecommunication systems, computer network and production systems can be formulated as finite queues with setup time and state dependent arrivals. This chapter deals with computational scheme to compute the exact stationary queue length distribution. In this chapter an efficient iterative algorithm is developed for computing the stationary queue length distribution in M/G/K/N queues with setup time and arbitrary state dependent arrival rates. The overall computation of the algorithm is O(N2) is complexicity. It can be of great use in application since it is easy to implement fast and quite accurate.

A generalized Conway–Maxwell–Poisson distribution which includes the negative binomial distribution

The Conway–Maxwell–Poisson (COM-Poisson) distribution with two parameters was originally developed as a solution to handling queueing systems with state-dependent arrival or service rates. This distribution generalizes the Poisson distribution by adding a parameter to model over-dispersion and under-dispersion and includes the geometric distribution as a special case and the Bernoulli distribution as a limiting case. In this paper, we propose a generalized COM-Poisson (GCOM-Poisson) distribution with three parameters, which includes the negative binomial distribution as a special case, and can become a longer-tailed model than the COM-Poisson distribution. The new parameter plays the role of controlling length of tail. The GCOM-Poisson distribution can become a bimodal distribution where one of the modes is at zero and is applicable to count data with excess zeros. Estimation methods are also discussed for the GCOM-Poisson distribution.

Tail asymptotics of a Markov-modulated infinite-server queue

This paper analyzes large deviation probabilities related to the number of customers in a Markov-modulated infinite-server queue, with state-dependent arrival and service rates. Two specific scalings are studied: in the first, just the arrival rates are linearly scaled by $$N$$ N (for large $$N$$ N ), whereas in the second in addition the Markovian background process is sped up by a factor $$N^{1+\varepsilon }$$ N 1 + ? , for some $$\varepsilon >0$$ ? > 0 . In both regimes (transient and stationary) tail probabilities decay essentially exponentially, where the associated decay rate corresponds to that of the probability that the sample mean of i.i.d. Poisson random variables attains an atypical value.

Non-Markovian bulk queueing system with state dependent arrivals and multiple vacations - a simulation approach

We considered a non-Markovian bulk arrival general bulk service queueing system with state dependent arrivals and multiple vacations in this paper. The arrivals occur in bulk with a rateλ, when the server is busy and with a rateλ0, when the server is on vacation. Using a supplementary variable technique and difference differential equations method we derive the queue size distribution, expected length of idle and busy periods and expected queue size of a Mx/G(a, b)/1 queueing system with state dependent arrivals and multiple vacations. We study the same through simulation using ARENA, a powerful manufacturing system simulation software and obtained some performance measures, which are mandatory for the study and analysis of the said queueing model. The simulated results for a few cases are presented.

Transient behavior of / /1 [ ] M G X Queueing Model With State Dependent Arrival, Balking and Bernoulli Vacation

This paper analyzes single server queueing system wherein the arrival units are in batches following compound Poisson process with varying arrival rates for dierent states. The arrival rates dier for dierent states like idle state, busy state and vacation period. The server provides service one by one and after completion of a service, the server either takes a vacation of random length with probability p (0 p 1) or continue to serve the next customer with probability 1-p, if any. Both service time and vacation time follow general distribution. In addition to this one of the customer impatience, balking is also discussed. We have obtained state dependent solution for the model the steady state analysis.

Analysis of unreliable bulk queue with statedependent arrivals

In this paper, we investigate a single-server Poisson input queueing model, wherein arrivals of units are in bulk. The arrival rate of the units is state dependent, and service time is arbitrary distributed. It is also assumed that the system is subject to breakdown, and the failed server immediately joins the repair facility which takes constant duration to repair the server. By using supplementary variable technique, we obtain the probability generating function of the number of units in the system which is further used to establish some performance indices such as the mean number of units in the system, mean waiting time, etc. Special cases are also discussed. In order to obtain approximate values of system state probabilities, the principle of maximum entropy is employed. Numerical results are also presented to validate the analytical formulae.

Analysis of M/G/ 1 queueing model with state dependent arrival and vacation

This investigation deals with single server queueing system wherein the arrival of the units follow Poisson process with varying arrival rates in different states and the service time of the units is arbitrary (general) distributed. The server may take a vacation of a fixed duration or may continue to be available in the system for next service. Using the probability argument, we construct the set of steady state equations by introducing the supplementary variable corresponding to elapsed service time. Then, we obtain the probability generating function of the units present in the system. Various performance indices, such as expected number of units in the queue and in the system, average waiting time, etc., are obtained explicitly. Some special cases are also deduced by setting the appropriate parameter values. The numerical illustrations are provided to carry out the sensitivity analysis in order to explore the effect of different parameters on the system performance measures.

Effect of application-layer rate-control mechanism on video quality for streaming services

In video streaming, feedback-based rate control is utilized for guaranteeing video quality on an end-to-end basis. This paper considers the effect of feedback-based rate controls on video quality from queueing theoretical point of view. We focus on a video streaming mechanism with a feedback-based rate control where a video server regulates the packet-sending rate according to the number of data blocks stored in a client node. The client node has two buffers: a synchronization buffer and a receiver buffer. Packets arriving to the client node are stored in the synchronization buffer first, and a video-data block is retrieved from a fixed number of packets in the synchronization buffer and forwarded to the receiver buffer. We model the client node as a discrete-time two-queue concatenated system with state-dependent packet arrivals, deriving the starvation and overflow probabilities. Numerical examples show the effectiveness of the feedback-based rate controls for improving both these probabilities. In particular, the rate control which is not sensitive to the number of data blocks in the receiver buffer makes these probabilities significantly small.

Queuing system with state-dependent controlled batch arrivals and server under maintenance

This paper deals with a single server Markovian queue subject to maintenance of the server. A batch of customers is allowed whenever the server is idle such that each individual customer in the batch is subject to a control admission policy upon arrival. Explicit expressions are obtained for the time dependent probabilities of the system size in terms of the modified Bessel functions. The steady state analysis and key performance measures of the system are also studied. Finally, some numerical illustrations are presented.

A preemptive resume priority retrial queue with state dependent arrivals, unreliable server and negative customers

In this paper we consider an unreliable single server retrial queue accepting two types of customers, with negative arrivals, preemptive resume priorities and vacations. A distinguishing feature of the model is that the rates of the Poisson arrival process depends on the server state. For this model we investigate the stability conditions and the joint queue length distribution in steady state. We also prove that our model satisfies the stochastic decomposition property. Transient, as well as steady state solutions for reliability measures are obtained. Finally, numerical results demonstrate the typical features of the model under consideration.

On moment stability properties for a class of state-dependent stochastic networks

We consider a class of stochastic networks with state-dependent arrival and service rates. The state dependency is described via multi-dimensional birth/death processes, where the birth/death rates are dependent upon the current population size in the system. Under the uniform (in state) stability condition, we establish several moment stability properties of the system: (i) the existence of a moment generating function in a neighborhood of zero, with respect to the unique invariant measure of the state process; (ii) the convergence of the expected value of unbounded functionals of the state process to the expectation under the invariant measure, at an exponential rate; (iii) uniform (in time and initial condition) estimates on exponential moments of the process; (iv) growth estimates of polynomial moments of the process as a function of the initial conditions. Our approach provides elementary proofs of these stability properties without resorting to the convergence of the scaled process to a stable fluid limit model.

V-uniform ergodicity for state-dependent single class queueing networks

We consider single class queueing networks with state-dependent arrival and service rates. Under the uniform (in state) stability condition, it is shown that the queue length process is V-uniformly ergodic; that is, it has a transition probability kernel which converges to its limit geometrically quickly in the V-norm sense. Among several asymptotic properties of V-uniformly ergodic processes, we present a Strassen-type functional law of the iterated logarithm result.

Performance Analysis of a Block-Structured Discrete-Time Retrial Queue with State-Dependent Arrivals

In this paper, we introduce a new discrete block state-dependent arrival (D-BSDA) distribution which provides fresh insights leading to a successful generalization of the discrete-time Markovian arrival process (D-MAP). The D-BSDA distribution is related to structured Markov chains and the method of stages. The consideration of this new discrete-time state-dependent block description gives one the ability of construct new stochastic models. The retrial queue analyzed in this paper gives an example of application of the D-BSDA distribution to construct more general and sophisticated models. We assume that the primary arrivals and the retrials follow the D-BSDA description and the service times are of discrete phase-type (PH). We study the underlying level dependent Markov chain of M/G/1-type at the epochs immediately after the slot boundaries. To this end, we employ the UL-type RG-factorization which provides an expression for the stationary probabilities. We also perform an analysis of waiting times. Numerical experiments are presented to study the system performance.

Transient analysis of a Markovian queue with chain sequence rates and total catastrophes

Queueing systems with state-dependent arrival and service rates along with catastrophes are investigated. Explicit expressions for the transient probabilities of system size are obtained. The corresponding steady-state probabilities are deduced. Busy period distribution is also found. Some performance measures are obtained for the system. Extensive numerical illustrations are provided. It is observed that the limiting distributions do not exist in the absence of catastrophes.

Transient Analysis for State-Dependent Queues with Catastrophes

Queueing systems with catastrophes and state-dependent arrival and service rates are considered. For two types of queueing systems namely, queues with discouraged arrivals and infinite server queue, explicit expressions for the transient probabilities of system size are obtained by using continued fractions technique. Some system performance measures and steady-state probabilities are studied. The effect of system parameters on system size probabilities are also illustrated numerically. It is observed that the steady-state probabilities differ when catastrophes are present, while they are identical in the absence of catastrophes.

Sequential settlement and site dependence in a migratory raptor

Habitat selection models, such as the ideal free, ideal despotic, site-dependent, and conspecific cuing models, are of great importance to behavioral ecologists given their capability to predict habitat distributions and to link individual behavior to population processes. However, there have been relatively few field tests of their predictions. We tested the 4 models by studying the process of sequential settlement on territory in 2 distant populations of a migratory raptor, the black kite Milvus migrans. Results were mainly consistent with the site-dependent model: on arrival, kites settled on progressively lower quality territories, and earlier arriving individuals were older, larger, and in better body condition than later arriving ones, leading to a state-dependent arrival sequence also predicted by a previous theoretical model of settlement pattern. Occupation of superior territories by superior phenotypes resulted in cascading advantages for earlier arriving individuals in terms of subsequent reproductive performance. At the population level, the populations expanded/retracted from lower quality sites during population increases/declines. The above scenario was consistent across the 2 populations, and a review of the literature uncovered a remarkably consistent picture of state-dependent arrival, progressive monopolization of best quality sites, and cascading effects on subsequent breeding performance. We propose as a general paradigm of sequential settlement the following process: 1) arrival date is a reliable surrogate of phenotype quality, 2) early-arriving individuals have preferential access to the best quality sites and partners, 3) the above conditions cascade into a number of benefits ultimately related to higher fitness for earlier arriving individuals.