\nPurpose\nThe authors consider parallel four-state tandem open queueing network. The queue capacity is infinite. Passenger arrival rate is Poisson distribution and service rate is exponential distribution. The queue is constructed in the form of tandem queue, and each and every queue of tandem queue is single server (M/M/1) queue. In tandem queue, passengers will leave the system once they receive service from both the states. The purpose of this paper is to provide performance analysis for four-state tandem open queue network, and a governing equation is formulated with the help of transition diagram. Using Burke theorem, the authors formulated equation for average number of passenger in the system, average waiting time of passenger in the system, average number of passenger in the queue and average waiting time of passenger in the queue.\n\n\nDesign/methodology/approach\nThis paper used Burke’s theorem.\n\n\nFindings\nIn this paper, performance analysis is done for parallel four-state tandem open queueing network and performance measure solved using Burkes theorem formula. K. Sreekanth et al. has done performance analysis for single tandem queue with three states. In this paper, the authors have done performance analysis for two tandem queues parallel with four states. This four-state tandem open queueing network is suitable for real world applications. This paper can extend for more number of service states and multi-server states according to the application, and in such case, the authors have to prove and explain with numerical examples. This analysis is more useful for the applications such as airports, railway stations, bus-stands and banks.\n\n\nOriginality/value\nIn this paper, parallel four-state tandem open queueing network and performance measure has been solved using Burke’s theorem formula.\n

Performance measures of parallel tandem open queueing network

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

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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J. Durga Aparajitha

Papers

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

Parallel and series queueing model with state and time dependent service