Time-dependent System Size Probabilities Research Articles

Analysis of state-dependent discrete-time queue with system disaster

An explicit expression for time-dependent system size probabilities is obtained for the general state-dependent discrete-time queue with system disaster. Using generating function for thenth state transient probabilities, the underlying difference equation of system size probabilities are transformed into three-term recurrence relation which is then expressed as a continued fraction. The continued fractions are converted into formal power series which yield the time-dependent system size probabilities in closed form. Further, the busy period distribution is obtained for the considered model. As a special case, the system size probabilities and busy period distribution of Geo/Geo/1 queue are deduced. Finally, numerical illustrations are presented to visualize the system effect for various values of the parameters.

Analysis of an M/M/c queue with heterogeneous servers, balking and reneging

This paper analyses a heterogeneous multi-server queuing system with balking and reneging. In this system, when all the c servers are busy, an arriving customer decides either to join the queue with probability p or balk with probability 1 - p. The customers waiting in the queue become impatient due to the long wait for service. Therefore, each individual waiting customer activates an independent impatience timer such that the customer's service starts before his timer expires, he gets the service and leaves the system after the completion of service. Otherwise, he abandons the system and never returns. The time-dependent system size probabilities are derived explicitly using generating function. Further the time-dependent mean, variance, busy period distribution and steady-state probabilities are obtained. Finally, some numerical illustrations are presented.

An analysis of discrete-time Geo/Geo/1 queue with feedback, repair and disaster

A discrete-time queue with feedback subject to system disaster, server failures and repair is considered in this paper. The time-dependent system-size probabilities are obtained by using generating functions where the system of difference equations in two parameters namely time epoch (m) and number of customers (n) are transformed into another difference equation in terms of generating functions. The difference equations of generating functions satisfy a three-term recurrence relation which leads to continued fraction. System-size probabilities and some performance measures in steady-state are derived. Further, busy period distribution, reliability and availability are also obtained. Numerical illustrations are provided for different parameter values to see their effect on performance measures and to get more in-sight of the model behaviour.

An analysis of discrete-time Geo/Geo/1 queue with feedback, repair and disaster

A discrete-time queue with feedback subject to system disaster, server failures and repair is considered in this paper. The time-dependent system-size probabilities are obtained by using generating functions where the system of difference equations in two parameters namely time epoch (m) and number of customers (n) are transformed into another difference equation in terms of generating functions. The difference equations of generating functions satisfy a three-term recurrence relation which leads to continued fraction. System-size probabilities and some performance measures in steady-state are derived. Further, busy period distribution, reliability and availability are also obtained. Numerical illustrations are provided for different parameter values to see their effect on performance measures and to get more in-sight of the model behaviour.

Transient analysis of M/M/1 queue with working vacation, heterogeneous service and customers’ impatience

This paper studies the time-dependent behavior of a single server queueing model with slow service during single and multiple working vacations, and customers’ impatience due to slow service. The time-dependent system size probabilities for the underlying model are obtained in closed form. Further, time-dependent mean and variance are deduced. Finally, numerical illustrations are shown to visualize the system effect for various values of parameters.

Transient analysis of an M/M/1 queueing system subject to differentiated vacations

This paper presents the transient analysis of an M/M/1 queueing system wherein the server is subject to two types of vacation, namely: vacation after the busy period and vacation taken immediately after the server has just returned from a previous vacation to find that there are no waiting customers. The arrivals are allowed to join the queue according to a Poisson distribution and the service takes place according to an exponential distribution. Whenever the system is empty, the server goes for a vacation and returns to the system after some random duration. If the server finds no waiting customer, he immediately takes another vacation of shorter duration. This paper presents explicit expressions for the time-dependent system size probabilities in terms of modified Bessel function of the first kind using generating function and Laplace transform techniques. Certain interesting performance measures like the mean and variance of the number of customers in the system, probability of the server to be in a busy or vacation state at an arbitrary time are explicitly obtained. Numerical illustrations are added to support the theoretical results.

Stationary and transient analysis of M/M/1/N queue subject to disasters and subsequent repair

A single server Markovian queuing model with finite capacity subject to disaster is considered. When the system is met with disasters, all customers (including those waiting) are lost forever and the system transits to the repair state. However, the customers are allowed to join the queue even when the system is under repair, but at a slower arrival rate. Repair times are also assumed to follow exponential distribution. In this paper, we obtain the stationary and time dependent system size probabilities of such a model using Laplace transform and generating function techniques. Certain performance measures of interest are also obtained in closed form.

Stationary and transient analysis of M/M/1/N queue subject to disasters and subsequent repair

A single server Markovian queuing model with finite capacity subject to disaster is considered. When the system is met with disasters, all customers (including those waiting) are lost forever and the system transits to the repair state. However, the customers are allowed to join the queue even when the system is under repair, but at a slower arrival rate. Repair times are also assumed to follow exponential distribution. In this paper, we obtain the stationary and time dependent system size probabilities of such a model using Laplace transform and generating function techniques. Certain performance measures of interest are also obtained in closed form.

Transient analysis of a single server discrete-time queue with system disaster

A discrete-time Geo/Geo/1 queue with system disaster is considered in this paper. The time-dependent and steady state probabilities of number of customers present in the system are obtained in terms of ballot numbers by solving the underlying system of difference equations using the generating function and continued fractions. Further, the busy period distribution is derived in terms of Catalan numbers. For special cases, time-dependent system size probabilities and busy period distribution are verified with the existing results in the literature. Numerical illustrations are provided for different parameter values to see their effect on performance measures and to get more insight of the model behavior.

Transient analysis of a two-heterogeneous servers queue with system disaster, server repair and customers’ impatience

A two-heterogeneous servers queue with system disaster, server failure and repair is considered. In addition, the customers become impatient when the system is down. The customers arrive according to a Poisson process and service time follows exponential distribution. Each customer requires exactly one server for its service and the customers select the servers on fastest server first basis. Explicit expressions are derived for the time-dependent system size probabilities in terms of the modified Bessel function, by employing the generating function along with continued fraction and the identity of the confluent hypergeometric function. Further, the steady-state probabilities of the number of customers in the system are deduced and finally some important performance measures are obtained.

Transient Analysis of M/M/1 Queue with Server Vacation Customers Impatient and a Waiting Server Timer

This paper studies an M/M/1 queuing model with server vacation and customers impatient in which the server activates a timer T whenever he returns from his vacation to an empty system and waits idle. This type of situation occurs most commonly in many real life situations as a human behavior. The time-dependent system size probabilities are derived explicitly in terms of modified Bessel functions and confluent hyper geometric functions. Also the time-dependent mean and variance are obtained analytically. Finally, some numerical illustrations are presented.

Transient Analysis of an M/M/1 queue with Multiple Exponential Vacation and N-Policy

A single server Markovian queueing model is considered. The arrivals are allowed to join the queue according to a Poisson distribution and the service takes place according to an exponential distribution. Whenever the system is empty, the server goes for a vacation and return back to the system after N or more customers are found in the system. If the number of customers in the system is less than ‘ ’ then the server takes another vacation. In this paper, we obtain explicit expressions for the time dependent system size probabilities of such a model using Laplace transform and generating function techniques. Numerical illustrations are added to support the theoretical results obtained.

Stationary and transient solution of Markovian queues - an alternate approach

This paper aims at presenting an alternate approach to derive the exact transient solution of Markovian queues. The continued fraction (Jacobi fraction) is expressed as power series and the power series coefficients are connected by a two-dimensional recurrence relation associated with infinite Stieltjes matrix equation. The recurrence relation is solved by using generating functions. A novel state-dependent birth-death queueing model is taken and expressed as continued fractions by employing integral transforms. The stationary probabilities of general state-dependent Markovian queues are obtained from the continued fractions and its associated tridiagonal determinants. As a special case the time-dependent system size probabilities and busy period distribution of a classical single server queue are deduced using continued fraction and its power series. Numerical illustrations are also presented.

Transient analysis of a queue with system disasters and customer impatience

A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate, resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter ?. If the system is not repaired before the customer's timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions.

A state-dependent queue alternating between arrivals and services

A queueing system usually involves an arrival process, service time distribution, service discipline, system capacity, etc. and steady-state performance measures extensively studied. However, time-dependent probabilities are important in applications and they do not involve convergence conditions. Further, several different models can have the same steady-state probabilities. We consider a novel state-dependent queueing system which alternating between arrival and service-states. We assume that the arrival, service and transfer rates depend on the number present in the system. Time-dependent system size probabilities and the duration of the busy period are obtained in a closed form for state-independent rates. We present numerical illustrations when the parameters are state-dependent. We have employed Continued Fractions effectively to achieve these transient results for this complex system.

Time-dependent analysis of a single-server retrial queue with state-dependent rates

For a single-server retrial queue with state-dependent exponential interarrival, service and inter-retrial times, we study the time-dependent system size probabilities by employing continued fractions and numerical illustrations are presented.

Exact Transient Solution of a Discrete Time Queue with State-Dependent Rates

SYNOPTIC ABSTRACTExact expressions are obtained in closed form for the time-dependent system size probabilities of a discrete queue with state-dependent arrival and service rates. The underlying difference equations of system size probabilities is converted into a continued fraction which is then expressed as a power series. As a special case we derive a Geo/Geo/1 queue. Numerical illustrations are presented.

Transient solution of a link with finite capacity supporting multiple streams

Most of the real-time applications involve connection establishment in point-to-point (unicast) communication in computer networks that can be modelled as a problem of resource allocation and resource sharing. These applications have different streams, such as video, voice, graphics, etc., each having different arrival rates. A stream (call) will be admitted into the network only if the network guarantees the ‘Quality of Service’ (QoS) parameters of the call such as bandwidth, delay, jitter and loss probability. Otherwise, the call is rejected. Hence, the study of call blocking probabilities in such communication networks is a problem of growing importance. Application of queueing theory to solve these problems has received considerable attention in the research community. Due to the real-time nature of the applications, it is pertinent to study the time-dependent behaviour of such systems. In this paper, a real-time unicast communication has been modelled as a multi-dimensional Markov process to obtain the time-dependent blocking probabilities. The time-dependent system size probabilities, means, variances and correlation coefficients of different streams are also obtained.

Transient Solution of a two-processor heterogeneous systems

A two-processor system heterogeneous with Poisson arrival of jobs having exponentially distributed execution times is considered. Here, the service rates of these processors are not identical. Each job requires exactly one processors for its execution and the scheduling policy is FCFS. When both the processors are idle, the faster processor is scheduled for service before the slower one. For this system, exact time-dependent system size probabilities are obtained using a suitable probability generating function. Finally, some important performance measures are also obtained.

A Birth and Death Process Related to the Rogers–Ramanujan Continued Fraction

Time-dependent system size probabilities of a birth and death process related to the Rogers–Ramanujan continued fraction are obtained. The range for the parameter in this continued fraction is obtained to ensure the positivity of the recursively defined birth and death rates. The general behavior of the birth and death rates is described and the asymptotic behavior of the transition probabilities and the (quasi)-stationary distribution is determined. For the transient case the birth and death process can be seen as a model in queuing theory where the length of the queue encourages customers to join the queue (the more the merrier).