Srinivas R. Chakravarthy

Analysis of a multi-server retrial queue with search of customers from the orbit

We consider a multi-server retrial queueing model in which customers arrive according to a Markovian arrival process (MAP). An arriving customer finding a free server enters into service immediately; otherwise the customer enters into an orbit of infinite size. An orbiting customer competes for service by sending out signals at random times until a free server is captured. The inter-retrial times are exponentially distributed with intensity depending on the number of customers in the orbit. Upon completion of a service, with a certain probability the server searches for an orbiting customer. Assuming the search time to be negligible, and the service and retrial times to be exponentially distributed, we perform the steady state analysis of the model using direct truncation and matrix-geometric approximation. Efficient algorithms for computing various steady state performance measures and illustrative numerical examples are presented.

A Multi-Server Retrial Queue with BMAP Arrivals and Group Services

In this paper, we consider a c-server queuing model in which customers arrive according to a batch Markovian arrival process (BMAP). These customers are served in groups of varying sizes ranging from a predetermined value L through a maximum size, K. The service times are exponentially distributed. Any customer not entering into service immediately orbit in an infinite space. These orbiting customers compete for service by sending out signals that are exponentially distributed with parameter θ. Under a full access policy freed servers offer services to orbiting customers in groups of varying sizes. This multi-server retrial queue under the full access policy is a QBD process and the steady state analysis of the model is performed by exploiting the structure of the coefficient matrices. Some interesting numerical examples are discussed.

Analysis of a finite MAP/G/1 queue with group services

The finite capacity queues, GI/PH/1/N and PH/G/1/N, in which customers are served in groups of varying sizes were recently introduced and studied in detail by the author. In this paper we consider a finite capacity queue in which arrivals are governed by a particular Markov renewal process, called a Markovian arrival process (MAP). With general service times and with the same type of service rule, we study this finite capacity queueing model in detail by obtaining explicit expressions for (a) the steady-state queue length densities at arrivals, at departures and at arbitrary time points, (b) the probability distributions of the busy period and the idle period of the server and (c) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures when services are of phase type are discussed. An illustrative numerical example is presented.

Analysis of a retrial queuing model with MAP arrivals and two types of customers

In this paper, we consider a queuing model in which two types of customers, say, Types 1 and 2, arrive according to a Markovian arrival process (MAP). Type 1 customers have a buffer of capacity K and are served in groups of varying sizes ranging from a predetermined value L to a maximum size, K. The service times are exponentially distributed. Type 2 customers are served one at a time and the service times are assumed to be exponential. The system has two servers, of which one is totally dedicated to serving Type 2 customers. The other server can serve both Types 1 and 2 customers. Any arriving Type 1 customer finding the buffer full is considered lost. Any Type 2 customer not entering into service immediately orbit in an infinite space. These orbiting customers compete for service by sending out signals that are exponentially distributed. The steady state probability vector of this queuing model is of matrix-geometric type with a highly sparse rate matrix. This sparsity is exploited in the analysis and several interesting numerical examples are discussed. The Laplace-Stieltjes transform LST of the waiting time distribution of a Type 1 customer at an arrival epoch is derived.

A disaster queue with Markovian arrivals and impatient customers

We consider a single server queueing system in which arrivals occur according to a Markovian arrival process. The system is subject to disastrous failures at which times all customers in the system are lost. Arrivals occurring during the time the system undergoes repair are stored in a buffer of finite capacity. These customers can become impatient after waiting a random amount of time and leave the system. However, these customers do not become impatient once the system becomes operable. When the system is operable, there is no limit on the number of customers who can be admitted. The structure of this queueing model is of GI/M/1-type that has been extensively studied by Neuts and others. The model is analyzed in steady state by exploiting the special nature of this type queueing model. A number of useful performance measures along with some illustrative examples are reported.

A Markovian inventory system with random shelf time and back orders

In this paper we consider an inventory system in which the demands occur according to a Markovian arrival process (MAP). The items in the inventory have shelf times that are assumed to follow an exponential distribution. The inventory is replenished according to an (s, S) policy and the replenishing times are assumed to follow a phase type distribution. Back orders of demands are allowed up to a certain level. Using matrix analytic methods we perform the steady state analysis of the inventory model. We also show that the reorder process can be modelled as a MAP. An optimization problem is proposed and several interesting numerical examples are presented.

Analysis Of A Multi-Server Queue With Markovian Arrivals And Synchronous Phase Type Vacations

We study a MAP/M/c queueing system in which a group of servers take a simultaneous phase type vacation. The queueing model is studied as a QBD process. The steady-state analysis of the model including the waiting time distribution is presented. Interesting numerical results are discussed.

A single server queue with platooned arrivals and phase type services

Abstract A semi-Markovian point process which qualitatively models platooned arrivals is introduced. This process is used as the input to a single server queue in which the service times are independent and have a common distribution of phase type. It is shown that this queue has an embedded Markov chain of a particular block-partitioned type, whose invariant probability vector in the stable case is of matrix-geometric form. Detailed algorithms for the computation of the steady-state features of the queue are obtained and a representative numerical example is discussed.

A k-out-of-n reliability system with an unreliable server and phase type repairs and services: the (N, T) policy

In this paper we study a k-out-of-n reliability system in which a single unreliable server maintains n identical components. The reliability system is studied under the (N, T) policy. An idle server takes a vacation for a random amount of time T and then attends to any failed component waiting in line upon completion of the vacation. The vacationing server is recalled instantaneously upon the failure of the N th component. The failure times of the components are assumed to follow an exponential distribution. The server is subject to failure with failure times exponentially distributed. Repair times of the component, fixing times of the server, and vacationing times of the server are assumed to be of phase type. Using matrix-analytic methods we perform steady state analysis of this model. Time spent by a failed component in service, total time in the repair facility, vacation time of the server, non-vacation time of the server, and time until failure of the system are all shown to be of phase type. Several performance measures are evaluated. Illustrative numerical examples are presented.

Maintenance of a deteriorating single server system with Markovian arrivals and random shocks

We consider a single server queue in which the arrivals occur according to a Markovian arrival process. The system is subject to external shocks causing the server to deteriorate and possibly fail. The maintenance of the server is provided either as a preventive one or for a complete failure so as to bring back to normal. Under the assumptions of Poisson shocks, exponential services and exponential maintenance with rates depending on the state of the server, and a general (discrete) probability distribution for the intensity of the shocks, the model is analyzed in steady-state. Some interesting theoretical results along with a few illustrative numerical examples are reported. An optimization problem involving various costs is studied numerically.