Kailash C. Madan

An M/G/1 queue with second optional service

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate λ (>0) all demand the first ‘essential’ service, whereas only some of them demand the second ‘optional’ service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/μ2 (μ2>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/Ek/1 and M/M/1 have been derived as particular cases.

A two-stage batch arrival queueing system with a modified bernoulli schedule vacation under N-policy

We consider a batch arrival queueing system, where the server provides two stages of heterogeneous service with a modified Bernoulli schedule under N-policy. The server remains idle till the queue size becomes N (>= 1). As soon as the queue size becomes at least N, the server instantly starts working and provides two stages of service in succession to each customer, i.e., the first stage service followed by the second stage service. However, after the second stage service, the server may take a vacation or decide to stay in the system to provide service to the next customer, if any. We derive the queue size distribution at a random epoch as well as a departure epoch under the steady state conditions. Further, we demonstrate the existence of the stochastic decomposition property to show that the departure point queue size distribution of this model can be decomposed into the distributions of three independent random variables. We also derive some important performance measures of this model. Finally, we develop a simple procedure to obtain optimal stationary operating policy under a suitable linear cost structure.

A two phase batch arrival queueing system with a vacation time under Bernoulli schedule

We consider a batch arrival queueing system, where the server provides two phases of heterogeneous service one after the other to the arriving batches under Bernoulli schedule vacation. After completion of both phases of service the server either goes for a vacation with probability r(0=

On a single server queue with two-stage heterogeneous service and deterministic server vacations

We analyse a single server queue with Poisson arrivals, two stages of heterogeneous service with different general (arbitrary) service time distributions and binomial schedule server vacations with deterministic (constant) vacation periods. After first-stage service the server must provide the second stage service. However, after the second stage service, he may take a vacation or may decide to stay on in the system. For convenience, we designate our model as M/G 1, G 2/D/1 queue. We obtain steady state probability generating function of the queue length for various states of the server. Results for some particular cases of interest such as M/Ek 1 , Ek 2 /D/1, M/M 1, M 2/D/1, M/E k /D/1 and M/G 1, G 2/1 have been obtained from the main results and some known results including M/Ek /1 and M/G/1 have been derived as particular cases of our particular cases.

On a single server queue with optional phase type server vacations based on exhaustive deterministic service and a single vacation policy

We analyze a single server queue with optional server vacations based on exhaustive service. Unlike other vacation policies, we assume that only at the completion of service of the last customer in the system, the server has the option to take a vacation or to remain idle in the system waiting for the next customer to arrive. The service times of the customers have been assumed to be deterministic and vacations are phase type exponential. We obtain explicit steady state results for the probability generating functions of the queue length, the expected number of customers in the queue and the expected waiting time of the customer. Some known results of the M/D/1 queue have been derived as a particular case.

On the Mx/G/1 queue with feedback and optional server vacations based on a single vacation policy

We study a single server queue with batch arrivals and general (arbitrary) service time distribution. The server provides service to customers, one by one, on a first come, first served basis. Just after completion of his service, a customer may leave the system or may opt to repeat his service, in which case this customer rejoins the queue. Further, just after completion of a customers service the server may take a vacation of random length or may opt to continue staying in the system to serve the next customer. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers and the average waiting time in the queue. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided.

A batch arrival Bernoulli vacation queue with a random setup time under restricted admissibility policy

We consider a batch arrival queue with a Bernoulli vacation bschedule, where, after completion of a service, the server either goes for a vacation of random length with probability θ(0 ≤ θ ≤ 1) or may continue to serve the next unit, if any, with probability (1−θ), under a Restricted Admissibility (RA) policy of arriving batches with a random Setup Time (SET). Unlike the usual batch arrival queueing system, the RA-policy differs during a busy period and a vacation period and hence all arriving batches are not allowed to join the system at all times. We derive the steady state queue size distributions at a random point of time as well as at a departure epoch. Also, we obtain some important performance measures of this model. Further, we demonstrate the existence of stochastic decomposition result for this type of model.

A two server queue with Bernoulli schedules and a single vacation policy

In this paper, we study a two server queue with Bernoulli schedules and a single vacation policy. We have assumed Poisson arrivals waiting in a single queue and two parallel servers who provide heterogeneous exponential service to customers on first-come, first-served basis. We consider two models, in one we assume that after completion of a service both servers can take a vacation while in the other we assume that only one may take a vacation. The vacation periods in both models are assumed to be exponential. We obtain steady state probability generating functions of system size for various states of the servers. A particular case is discussed and a numerical example is provided.

A single channel queue with bulk service subject to interruptions

Abstract This paper studies a single channel queueing system with Poisson arrivals and exponential service in batches of fixed size b ( ≥ 1). However, the service channel is subject to occasional breakdowns occuring randomly in time. Both the operative times and the repair times of the service channel have been assumed to be exponential. The probability generating function of the queue length has been obtained for the steady state. For a particular case b = 1, the steady state solutions as well as the average queue length have been obtained explicitly. Also some known steady state results have been derived in another particular case.

On MxG1G21 queue with optional re-service

We study a single server queue with batch arrivals and two types of heterogeneous service with different general (arbitrary) service time distributions. The server provides either type of service to customers, one by one, on a first come, first served basis. Just before a service starts, a customer has the option to choose either type of service after completion of which the customer may leave the system or may opt for re-service of the service taken by him. We obtain steady-state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue and the system, the average number of customers and the average waiting time in the queue as well as the system. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided.