A. Krishnamoorthy

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.

k -out-of- n system with repair: the max( N, T ) policy

We consider a k-out-of-n system with repair under the max(N, T) policy. Under this policy, the repair facility is activated for repair of failed units whenever the maximum of an exponentially distributed time duration T and the sum of N (1 ≤ N ≤ n - k) random variables is realized. The repair times and lifetimes of components are assumed to be independent exponentially distributed random variables. The repaired units are assumed to be as good as new. Failed units are repaired one at a time. The repair facility is switched off the moment all failed units are back to operation. System state probabilities in the long run are derived for (a) cold (b) warm and (c) hot systems. System reliability, the distribution of time during which the server is continuously engaged and its expected duration are computed. Several other system characteristics are also obtained. Determination of the optimal values of N and α is discussed and some numerical illustrations are provided.

k -out-of- n : G system with repair: the D -policy

Abstract In this paper we consider a k -out-of- n : G system with repair under D -policy. According to this policy whenever the workload exceeds a threshold D a server is called for repair and starts repair one at a time. He is sent back as soon as all the failed units are repaired. The repaired units are assumed to be as good as new. The repair time and failure time distributions are assumed to be exponential. We obtain the system state distribution, system reliability, expected length of time the server is continuously available, expected number of times the system is down in a cycle and several other measures of performance. We compute the optimal D value which maximizes a suitably defined cost function. Scope and purpose This paper considers a repair policy, called D -policy, of a k -out-of- n : G system. In a k -out-of- n : G system, the system functions as long as there are atleast k operational units. The server activation cost is high once it becomes idle due to all failed units repaired. Hence it is activated when the accumulated amount of work reaches D . This paper examines the optimal D -value by bringing in costs such as the cost of system being down, the server activation cost. Activating the server the moment the first failure takes place may involve very heavy fixed cost per cycle (a cycle is the duration from a point at which the server becomes idle to the next epoch at which it becomes idle after being activated). The other extreme of server activation only after n − k +1 units fail leads to the system being down for a long duration in each cycle. Hence the need for the optimal D -policy. A brief account of k -out-of- n : G system can be had in Ross (Ross, SM. Introduction to probability models, 6th ed., New York: Academic Press, 1997). The results obtained here find direct applications in reliability engineering, Production systems, Satellite communication, etc.

k-out-of-n-system with repair:T-policy

We consider a k-out-of-n system with repair underT-policy. Life time of each component is exponentially distributed with parameter λ. Server is called to the system after the elapse ofT time units since his departure after completion of repair of all failed units in the previous cycle or until accumulation ofn — k failed units, whichever occurs first. Service time is assumed to be exponential with rateμ.T is also exponentially distributed with parameter α. System state probabilities in finite time and long run are derived for (i) cold (ii) warm (iii) hot systems. Several characteristics of these systems are obtained. A control problem is also investigated and numerical illustrations are provided. It is proved that the expected profit to the system is concave in α and hence global maximum exists.

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.

Reliability of ak-out-of-n system with repair and retrial of failed units

We consider ak-out-of-n system with repair. Life times of components are independent exponentially distributed random variables with parameter λi when the number of working units isi. Failed units are taken for repair to a station, manned by a single server, having no waiting room. The failed units are brought to an orbit, if the server is found to be busy, for retrial. Reliability of the system is computed in the following three situations: (i) Cold system (ii) Warm system and (iii) Hot system. Several other system characteristics are derived.

Inventory with disaster

We Consider a continuous review (s,S) inventory system in which commodities are damaged due to decay and disaster. Demands for items follow Poisson process and the life times of an item and the times between the disasters exponential. Assuming no shortages and zero lead time, transient and steady state probabilities on inventory levels are derived. Optimization is discussed with numerical illustrations

A queueing system with single arrival bulk service and single departure

In this note, we study a Markovian queueing system with accessible batches for service, but units depart individually. This system generalizes the M/M/C, bulk service, and accessible batch queueing systems. We compute the system size probabilities in transient and steady states, waiting time distribution, busy period distribution, expected queue length, etc. Littles formula is shown to hold for the present set up. We investigate the optimal values of the maximum batch size for service and capacity of the system.

On a bulk arrival bulk service infinite service queue

This paper considers an infinite server queue in continuous time in which arrivals are in batches of variable size X and service is provided in groups of fixed size R. We obtain analytical results for the number of busy servers and waiting customers at arbitrary time points. For the number of busy servers, we obtain a recursive relation for the partial binomial moments both in transient and steady states. Special cases are also discussed

GI/M/1/1 queue with finite retrials and finite orbits

In this paper we consider the GI/M/1/1 retrial queue with finite number of retrials to each orbital customer and a finite number of orbits. The interarrival times from outside the system have a general distribution. The sojourn time of a unit in an orbit until its retrial for service and its service time are exponentially distributed with parameters depending on the orbit number. The maximum number of retrials any unit is permitted to take is restricted to k. There are a finite number, say m, of orbits with at most one customer in each orbit. At the time of an arrival if the server is busy and all orbits are occupied, then the customer is lost to the system. If the server is idle at an arrival epoch, the unit directly goes for service. If the server is busy and at least one of the orbits is empty then the arriving customer occupies the first empty orbit. A unit in the orbit retries for service which returns to the same orbit (if the server is busy) with probability P and leaves the system with probability 1−P, 0<P<1. We compute the system state distribution. The distribution of the number of customers reneging and that of the number of units served during an interarrival time are computed. Also waiting time distribution and the idle period distribution are computed. Optimal values of k and m are investigated.