The present investigation deals with the multi component machine repair problem by including the concepts of reneging and interdependent controlled rates. We consider a machining system consisting of M operating units, S 1 and S 2 warm spares of two kinds and a repair facility having R permanent repairmen and r additional removable repairmen. When all the spares are used and the operating units fail, the system works in degraded mode. By using interdependent controlled arrival and service rates and reneging parameters along with time-sharing concept, the queueing model is developed for more versatile situations encountered in production/ manufacturing industries. The solution is obtained by using recursive technique. The expressions for the queue size distribution and expected queue length are established. Some more performance indices such as expected number of failed units waiting for the service, expected number of busy repairmen, expected number of both types of spare parts functioning as standbys, etc., are obtained with the help of recursive technique. By taking numerical illustrations, the sensitivity analysis is provided to validate the analytical results.

M/M/R+r machining system with reneging, spares and interdependent controlled rates

This paper develops and analyzes a two node tandem queueing model with phase type service having time and state dependent service rates. Here, it is assumed that the service processes of the two service stations follow non-homogenous Poisson processes and service rates are dependent on the number of customers in the queue connected to it. Using the difference-differential equations, the joint probability generating function of the queue size distribution is derived. The system performance measures such as average number of customers in the queue, throughput of the service stations, and average waiting time of customers in the queue and in the system and the variance of the number of customers in each queue are derived. A numerical illustration is presented. The sensitivity analysis of the model revealed that the time and load dependent service rates have significant influence on congestion of queues and waiting time. The transient analysis can predict the performance measures more accurately for small period of time. This model can also include some of the early models as particular cases.

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Two Node Tandem Queueing Model with Phase Type State and Time Dependent Service Rates

This paper addresses the solution procedures for analyzing queuing systems with non-stationary arrival and service processes. Here, it is assumed two queues are connected in tandem and customers arrive to the first node following a non-homogeneous Poisson process. The service processes in both the nodes follow nonhomogeneous Poisson processes. It is further assumed that the customer after completing the service from the first service station may join the second queue or leave the system with certain probability. The service rates in both the service stations are dependent on the number of customers in the queue connected to them. The model behavior is analyzed under transient condition by deriving explicit expressions for the average number of customers in each queue, the waiting time of customers in each queue, the throughput of the nodes, the variability of the number of customers in each queue, the emptiness of the queues. The sensitivity analysis of the model reveals that the nonhomogeneous arrival and service processes can analyze the behavior of the queuing system more accurately under transient conditions. The load dependent service strategy can reduce congestion in queues and avoid delay in transmission. This model also includes some of the earlier queuing models as particular cases.

Two Node Tandem Queuing Model with Phase Type Load Dependent Service Having Non-Homogeneous Arrival and Service Processes

In this paper, we develop and analyze an interdependent forked queueing model with state dependent service times. Here, it is assumed that the arrival and service processes are correlated and follows a multivariate poisson process. Using the differencedifferential equations, the joint probability generating function of the number of customers in each queue is derived. The system performance like the average number of customers in each queue, the average waiting time of a customer, the throughput of each service station, the idleness of the servers are derived explicitly. The sensitivity analysis of the model reveals that the dependency parameter and state dependent service rates can reduce congestion in queues and average waiting time of the customer. This model also includes some of the earlier models as particular cases for specific values of the parameters. The forked queueing models are much useful for analyzing and monitoring several communication networks and production processes.

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Kumar K.Naveen

Papers

Transient Analysis of an Interdependent Forked Tandem Queuing Model with Load Dependent Service Rate