Stationary tail asymptotics of a tandem queue with feedback

Abstract

Motivated by applications in manufacturing systems and computer networks, in this paper, we consider a tandem queue with feedback. In this model, the i.i.d. interarrival times and the i.i.d. service times are both exponential and independent. Upon completion of a service at the second station, the customer either leaves the system with probability p or goes back, together with all customers currently waiting in the second queue, to the first queue with probability 1 − p. For any fixed number of customers in one queue (either queue 1 or queue 2), using newly developed methods we study properties of the exactly geometric tail asymptotics as the number of customers in the other queue increases to infinity. We hope that this work can serve as a demonstration of how to deal with a block generating function of GI/M/1 type, and an illustration of how the boundary behaviour can affect the tail decay rate.

Performance Evaluation of Three-Node Tandem Communication Network Model with Feedback for First Two nodes Having Non Homogeneous Poisson Arrivals

“…Different values of the parameters are taken for bandwidth allocation and arrival of packets. The packet arrival (λ) varies from 2x10 4 packets/sec to 7x10 4 packets/sec, probability parameters (θ,) varies from 0.1 to 0.9, the transmission rate for first transmitter (µ 1 ) varies from 5x10 4 packets/sec to 9x10 4 packets/sec, transmission rate for second transmitter (µ 2 ) varies from 15x10 4 packets/sec to 19x10 4 packets/sec and transmission rate for third transmitter (µ 3 ) varies from 25x10 4 packets/sec to 29x10 4 packets/sec. Dynamic Bandwidth Allocation strategy is considered for both the two transmitters.…”

“…It is also observed from the table 4.1 that, as the time (t) increases, the throughput of first, second and third transmitters is increasing for the fixed values of other parameters. When the parameter λ increases from 3x10 4 packets/sec to 7x10 4 packets/sec, the throughput of three transmitters increases. As the parameter increases, the throughput of transmitters also increases.…”

“…As the second probability parameter  value increases from 0.1 to 0.9, the throughput of the first transmitter remains constant, throughput of the second transmitter increases and throughput of the third transmitter decreases. As the transmission rate of the first transmitter (µ 1 ) increases from 5x10 4 packets/sec to 9x10 4 packets/sec, the throughput of the first, second and third transmitters increases. The transmission rate of the second transmitter (µ 2 ) increases from 15x10 4 It is observed from the Table 4.2 that as the time (t) varies from 0.1 to 0.9 seconds, the mean number of packets in the three buffers and in the network increases when other parameters are kept constant.…”

“…As the transmission rate of the first transmitter (µ 1 ) increases from 5x10 4 packets/sec to 9x10 4 packets/sec, the throughput of the first, second and third transmitters increases. The transmission rate of the second transmitter (µ 2 ) increases from 15x10 4 It is observed from the Table 4.2 that as the time (t) varies from 0.1 to 0.9 seconds, the mean number of packets in the three buffers and in the network increases when other parameters are kept constant. When the λ changes from 3x10 4 packets/second to 7x10 4 packets/second the mean number of packets in the first, second, third buffers and in the network are increasing.…”

“…Communication networks are an important application of such systems. Tang and Zhao, assumed Markovian arrival and service times for feedback queues [4]. On completion of a service at the station, the task either leaves the system with probability p, or goes back, along with all tasks currently waiting in the queue, to the station with probability 1 -p. Tandem queues with feedback have comprehensively studied in the literature [5].…”