Balasubramanian Krishna Kumar

Balasubramanian Krishna Kumar
Anna University, Chennai · Department of Mathematics

M.Sc., M.Phil., Ph.D.

About

84
Publications
23,733
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1,121
Citations
Additional affiliations
January 2000 - December 2012
Anna University, Chennai
January 1991 - December 1993
Indian Institute of Technology Madras

Publications

Publications (84)
Article
Full-text available
This paper studies a Markov model of a queuing-inventory system with primary, retrial, and feedback customers. Primary customers form a Poisson flow, and if an inventory level is positive upon their arrival, they instantly receive the items. If the inventory level is equal to zero upon arrival of a primary customer, then this customer, according to...
Article
Full-text available
We analyze a multi-processor two-stage tandem call center retrial queueing network in which the processors are subject to active breakdowns and repairs at stage-I. A level-dependent quasi-birth-and-death (LDQBD) process is formulated and a sufficient condition for ergodicity of the system is discussed. Under the stability condition, the stationary...
Article
Full-text available
An M/G/1 retrial queueing system with two phases of service of which the second phase is optional and the server operating under Bernoulli vacation schedule is investigated. Further, the customer is allowed to balk upon arrival if he finds the server unavailable to serve his request immediately. The joint generating functions of orbit size and serv...
Preprint
We investigate an M/M/1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of t...
Article
We investigate an M/M/1 queue operating in two switching environments, where the switch is governed by a two-state time-homogeneous Markov chain. This model allows to describe a system that is subject to regular operating phases alternating with anomalous working phases or random repairing periods. We first obtain the steady-state distribution of t...
Article
Full-text available
Multiprogramming and Multiporgramming Retrial Queue with Berenoulli Vacation of CPU servers is discussed
Conference Paper
We consider the transmission of protocol data units in an LTE eNodeB in downlink direction and focus on the radio link control (RLC) and hybrid Automatic-Repeat-Request functionality at layer 2 of the transmission system. We model the associated window flow control of RLC frames in terms of an open queueing network with two stations and describe RL...
Article
Full-text available
This paper deals with retrial queueing models having an unlimited/a finite orbit capacity with control retrial policy of a multiprogramming–multiprocessor computer network system. Under the Markovian assumptions and light-traffic condition, the steady-state probabilities of the number of programs in the system and the mean number of programs in the...
Article
Full-text available
A single server queue subject to maintenance of the server and the close down period is considered. We obtain explicit expressions for the transient probabilities of the system size, the server under maintenance state and the close down period. The time-dependent performance measures of the system and the probability density function of the first-p...
Article
Full-text available
Sudhesh (2010) has discussed the transient probabilities for queueing systems subject to catastrophic failures and impatience of customers. However, it contains some errors concerning the terminology and the final forms of the transient probabilities of the system size. In this paper, we correct the errors and obtain the correct results in the work...
Article
This paper is concerned with the analysis of a single server queueing system subject to Bernoulli vacation schedules with server setup and close down periods. An explicit expression for the probability generating function of the number of customers present in the system is obtained by using imbedded Markov chain technique. The steady state probabil...
Article
Full-text available
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 te...
Article
Full-text available
A non-Markovian feedback single-server retrial queue with collisions and general retrial times is investigated. A necessary and sufficient condition for the system to be stable is studied. Using the supplementary variable technique, the joint distribution of the server state and the orbit length under steady-state is obtained. Some interesting and...
Article
Full-text available
A single server retrial queue with negative customers and two types of Bernoulli feedback is considered. A necessary and sufficient condition for the system to be stable is investigated. The system size probabilities at output epochs are obtained by using an embedded Markov chain. Further, the joint generating functions of queue length and server s...
Article
Full-text available
Consider a system performing a continuous-time random walk on the integers, subject to catastrophes occurring at constant rate, and followed by exponentially-distributed repair times. After any repair the system starts anew from state zero. We study both the transient and steady-state probability laws of the stochastic process that describes the st...
Article
A Markovian single server feedback retrial queue with linear retrial rate and collisions of customers is studied. Using generating function technique, the joint distribution of the server state and the orbit length under steady-state is investigated. Some interesting and important performance measures of the system are obtained. Finally, numerical...
Article
Full-text available
In this paper, a steady-state Markovian multi-server retrial queueing system with Bernoulli vacation scheduling service is studied. Using matrix-geometric approach, various interesting and important system performance measures are obtained. Further, the probability descriptors like ideal retrial and vain retrial are provided. Finally, extensive num...
Article
Full-text available
In this paper, a Markovian multiple server queue in which each server takes vacation according to Bernoulli scheduling service has been studied. For this system, the stationary queue length distribution and several performance characteristics are obtained using the matrix geometric solution technique. The system busy period and waiting time distrib...
Article
Full-text available
This paper deals with a multiserver feedback retrial queueing system with finite waiting position and constant retrial rate. This system is analyzed as a quasi-birth-and-death (QBD) process and the necessary and sufficient condition for stability of the system is investigated. Some important system performance measures are obtained using matrix geo...
Article
Full-text available
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...
Article
Full-text available
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 measur...
Article
Full-text available
We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is...
Article
Full-text available
A transient solution is obtained analytically using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and non-zero repair time. The steady state probability of the system size is present. Some key performance measures, namely, throughput, loss probability and response time for the system under co...
Article
Full-text available
This paper presents a transient solution for the system size in an M/M/2 queue where the service rates of the servers are not identical with the possibility of catastrophes in the system. The time dependent probabilities for the number in the system are obtained. The steady state probabilities of the system size are also provided. Some important pe...
Article
Full-text available
This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of a positive customer who initiates a busy period is G e (x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution G b (x)....
Article
Full-text available
This paper discusses multiserver feedback retrial queues with balking and control retrial rate. This system is analyzed as a quasi-birth-and-death (QBD) process and the necessary and sufficient condition for stability of the system is discussed. Some interesting system performance measures are obtained using matrix geometric method. The effects of...
Article
In this paper, a Markovian queue with two heterogeneous servers and multiple vacations has been studied. For this system, the stationary queue length distribution and mean system size have been obtained by using matrix geometric method. The busy period analysis of the system and mean waiting time distribution are discussed. Extensive numerical illu...
Article
Full-text available
A transient solution for the system size in the M/M/1 queueing system with the possibility of catastrophes and server failures is analyzed. The steady state probabilities of the system size are also derived. Some important performance measures are discussed. Finally, the reliability and availability of the system are obtained.
Article
Full-text available
An age-dependent branching process where disasters occur as a renewal process leading to annihilation or survival ofall the cells, is considered. For such a process, the total mean sojourn time of all the cells in the system is analysed using the regeneration point technique. The mean number of cells which die in timet and its asymptotic behaviour...
Article
Full-text available
This paper is concerned with the analysis of a single-server batch arrival retrial queue with two classes of customers. In the case of blocking, the class-1 customers leave the system forever whereas the class-2 customers leave the service area and enter the orbit and wait to be served later. The necessary and sufficient condition for the system to...
Article
Full-text available
This paper is concerned with the analysis of a single-server batch arrival retrial queue with Bernoulli service schedule and multiple vacation with general retrial times and the server being subject to starting failures. We assume that the customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served...
Article
An M/G/1 queue with Bernoulli feedback and general setup time under a control policy is analyzed. The customers arrival rate at service station varies according to the system status: buildup, setup and busy states. The probability generating function for system size and mean number of customers in the system are obtained under steady-state conditio...
Article
This paper discusses a retrial queue with Bernoulli feedback, where the server is subjected to starting failure. The retrial time is assumed to follow an arbitrary distribution and the customers in the orbit access the server under FCFS discipline. The necessary and sufficient condition for the stability of the system is derived. Various performanc...
Article
Full-text available
An M/G/1 retrial queueing system with additional phase of service and possible preemptive resume service discipline is considered. For an arbitrarily distributed retrial time distribution, the necessary and sufficient condition for the system stability is obtained, assuming that only the customer at the head of the orbit has priority access to the...
Article
An M/G/1 queueing system with no waiting room is considered. The server is assumed to provide two types of services-regular and optional with the provision of server breakdown. Explicit expressions for the transient probabilities for the system being idle, busy and breakdown are obtained. The reliability and availability analysis of the system are...
Article
This paper is concerned with the analysis of a single-server queue with Bernoulli vacation schedules and general retrial times. We assume that the customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed access to the server....
Article
Full-text available
This paper presents a transient solution for the system size in the M/M/2 queue with the possibility of catastrophes at the service stations. The state probability of the system size at time t, where the queue starts with any number of customers, in obtained. Asymptotic behaviour of the probability of the server being idle and mean system size are...
Article
Full-text available
This paper analyses a multi-channel Markovian queueing system with heterogeneous servers and balking behaviour. For the system initiated with a random number of customers, the transient solution for the system size distribution is obtained explicitly in a direct way and the steady state probabilities are deduced. Some special cases are also discuss...
Article
This paper discusses an M/G/1/1 queue where customer after regular service completion may leave the system or feedback to regular service again or opt for specialised services. Explicit expressions for the transient probabilities that the server is idle and the server is providing the regular service or optional services are obtained.
Article
A transient solution for the system size in the M/M/1 queueing model with the possibility of catastrophes at the service station is derived in the direct way. Asymptotic behavior of the probability of the server being idle and mean queue size are discussed. Steady-state probabilities are also obtained.
Article
A semi-Markov cornpartmental model with branching particies is considered. The notion of disaster is incorporated into the structure. The means of (i) the total sojourn time, (ii) the number of deaths, (iii)the number of births and (iv)the number of emigrant particles in the system are analysed. Some interesting relations connecting these means are...
Article
Full-text available
The total number of deaths and total sojourn times of African honey bees are studied using semi-markov compartment analysis. This generalizes many existing biological models.
Article
A multitype Markov branching process under the influence of disasters (catastrophes) arriving as a renewal process leading to possible mutation of particles to other types is considered. The integral equations for the probability generating function of the number of particles of each type are derived incorporating time dependent mutation rates. The...
Article
We consider a multitype Markov branching process under the influence of disasters occurring as a renewal process. For the branching particles in the system, the disasters may lead to possible death or mutation to other types of particles. For such a process, the total mean sojourn time of all the particles in the system is analysed using the regene...
Article
A semi-Markov multi-compart mental system in which particles reproduce similar particles as a Markov branching process and being subjected to disasters is studied. Expressions for the mean number of particles alive at time t in each compartment are obtained. The results concerning irreversible, mammillarian and catenary compartmental systems have b...
Article
The analysis for busy period of an infinite capacity M/M/1 queue with balking is discussed. Defining the generating function in an unusual and direct way, the time-dependent solution for the density function of busy period is obtained elegantly.
Article
In this paper we consider a discrete-time cyclic-service system consisting of multiple stations visited by a single server. Several priority classes of customers arrive at each station according to independent batch Bernoulli processes. The head-of-the-line (HL) priority rule and non-zero switch-over times between stations are assumed. The customer...
Article
We consider a stochastic compartmental model in which cells reproduce in accordance with a regulated birth and death process. We find expressions for the mean vector and covariance matrix for the number of cells in these compartments. We obtain the asymptotic behavior of the mean vector in the general case and explicit expressions for two compartme...
Article
We consider a discrete-time cyclic-service system consisting of multiple stations visited by a single server. Customers from several priority classes arrive at an individual station according to independent batch Bernoulli processes. We assume a non-preemptive priority rule and non-zero switch-over times of the server between consecutive stations....
Article
An infinite capacityM/M/1 queue with balking is discussed. Defining the generating function in an unusual and direct way, the time-dependent solution for the system size is obtained elegantly.
Article
A semi‐Markov compartmental system in which the particles reproduce according to the Markov branching process, apart from transitions between the compartments, is considered. Asymptotic behaviour of the mean matrix of the number of particles alive at time t is studied. Explicit expressions for some special cases are discussed.
Article
A density‐dependent birth and death process with immigration is considered. An explicit expression is obtained for the population size {X(t)}in terms of the modified Bessel functions. Asymptotic behaviours of the mean and P(X(t) =0) are discussed. A Martingale function is constructed and the probability of population becoming extinct before reachin...
Article
We consider a semi-Markov compartroental system with branching particles, Me study the total sojourn time of all the particles and obtain the asymptotic behaviour of its mean. We also discuss some special systems.
Article
A density–dependent birth and death process in which immigration is permitted only when the number of particles in the system is less than a positive integer kis considered. An explicit expression for the probability generating function of the population size is obtained by using a direct and unconventional approach. Special cases are discussed.
Article
A density-dependent birth and death process is considered, in which immigration is allowed whenever the population size reaches zero. An explicit expression is obtained for the population size in terms of the modified Bessel functions. Asymptotic behaviour of extinction probability and the mean are established.
Article
An age-dependent branching process where each cell produces offspring only if its life length is greater than τ1 > 0 is considered. Asymptotic behaviours of the mean number of cells that die after time τ2 > 0 for an age-dependent branching process are studied for the different cases and exact expressions for a Markov branching process are given. Su...
Article
An example of density dependent-birth and death process whose mean satisfies the logistic equation as proposed by Gompertz is given. Explicit expressions for the probability generating function and non-trivialstationarydistribution are obtained.
Article
Two examples of density dependent birth and death processes whose means satisfy the logistic equation as proposed by Verhulst and Gompertz are given. Explicit expressions for the probability generating functions and non-trivial stationary distributions are obtained.
Article
For an age-dependent branching process, we consider cells at time t whose lifetimes lie in an interval. The asymptotic behaviour of the mean of these cells is given and the sum of their lifetimes is discussed.

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Projects (2)
Project
This is a joint RFBR (Russia) - DST (India) project of an international collaboration group from Karelian Research Centre of RAS, Federal Research Centre "Computer Science and Control" of RAS from Russian side; Anna University Chennai, Mahindra Ecole Centrale from Indian side. High-Performance and Distributed Computing systems are widely used to equip science and business to perform computationally intensive applications. To fulfill this unsaturated demand, Ubiquitous Computing attempts to utilize idle resources of nonspecific computational resources enabled by modern telecommunication technologies such as Internet of Things. To guarantee high rate of data transmission in the high-performance infrastructure, various media are used at network level, including the wired/wireless transmission systems. In such systems the variable transmission quality is leveraged by means of Automatic Repeat Request technologies and Error Control. These technologies bring up issues of reliability, performance, efficiency and stability of such systems. Of specific interest is the energy efficiency issue, which is open for all the aforementioned system classes, despite their heterogeneity. Random and unpredictable characteristics of workload submit to necessity of stochastic models in the performance analysis of these systems. In the field of such models, all the aforementioned system classes are related to the so-called multiserver queueing systems, however possessing very specific features. Large scale of real-world applications and sophisticated system structure lead to dramatic increase of analysis complexity. A separate difficulty is related to transfer of the theoretical results towards practical applications. The proposed project is targeted to solve the problem of analysis and performance estimation of high-performance, distributed and ubiquitous computing systems, as well as modern telecommunication systems with automatic repeat request and error control. The new methodology of validated three level stochastic modeling (at analytical, simulation and technical levels) of stationary performance and transient behavior in transient and steady-state regimes is implemented in the project. At analytical level the queueing theory is applied, utilizing the enhanced matrix-analytic and regenerative methods (in particular, applied jointly). At the simulation level it is assumed to apply the discrete event simulation technique (based on the so-called piecewise deterministic Markov chains), regenerative simulation (including confidence estimation), applying the novel regenerative envelops method, variance reduciton techniques (e. g. antithetic variables) as well as stochastic recursions. Finally, the results of technical level modeling are expected from the mockup real-life experiments based on open source software and multicore devices, wireless network devices and appropriate measurement devices (for energy efficiency).
Project
To Get Performance Measures