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In this paper, we consider a queue whose service speed changes according to an external environment that is governed by a Markov process. It is possible that the server changes its service speed many times while serving a customer. We derive first and second moments of the service time of customers in system using first step analysis to obtain an insight on the service process. In fact, we obtain an intriguing result in that the moments of service time actually depend on the arrival process! We also show that the mean service rate is not the reciprocal of the mean service time.Further, since it is not possible to obtain a closed form expression for the queue length distribution, we use matrix geometric methods to compute performance measures such as average queue length and waiting time. We apply the method of large deviations to obtain tail distributions of the workload in the queue using the concept of effective bandwidth. We present two applications in computer systems: (1) Web server with multi-class requests and (2) CPU with multiple processes. We illustrate the analysis and various methods discussed with the help of numerical examples for the above two applications.
In this paper, we consider a queue whose service speed changes according to an external environment that is governed by a Markov process. It is possible that the server changes its service speed many times while serving a customer. We derive first and second moments of the service time of customers in system using first step analysis to obtain an insight on the service process. In fact, we obtain an intriguing result in that the moments of service time actually depend on the arrival process! We also show that the mean service rate is not the reciprocal of the mean service time.Further, since it is not possible to obtain a closed form expression for the queue length distribution, we use matrix geometric methods to compute performance measures such as average queue length and waiting time. We apply the method of large deviations to obtain tail distributions of the workload in the queue using the concept of effective bandwidth. We present two applications in computer systems: (1) Web server with multi-class requests and (2) CPU with multiple processes. We illustrate the analysis and various methods discussed with the help of numerical examples for the above two applications.
The Rate Control Protocol (RCP) uses explicit feedback from routers to control network congestion. RCP estimates it's fair rate from two forms of feedback: rate mismatch and queue size. An important design question that remains open in RCP is whether the presence of queue size feedback is helpful, given the presence of feedback from rate mismatch. The feedback from routers to endsystems is time delayed, and may introduce instabilities and complex non-linear dynamics. Delay dynamical systems are often modeled using delay differential equations to facilitate a mathematical analysis of their performance and dynamics. The RCP models with and without queue size feedback give rise to two distinct non-linear delay differential equations. Earlier work on this design question was based on methods of linear systems theory. For further progress it is quite natural to employ nonlinear techniques. In this study, we approach this design question using tools from control and bifurcation theory. The analytical results reveal that the removal of queue feedback could enhance both stability and convergence properties. Further, using Poincaré normal forms and center manifold theory, we investigate two nonlinear properties, namely, the type of Hopf bifurcation and the asymptotic stability of the bifurcating limit cycles. We show that the presence of queue feedback in the RCP can lead to a sub-critical Hopf bifurcation, which would give rise either to the onset of large amplitude limit cycles or to unstable limit cycles. Whereas, in the absence of queue feedback, the Hopf bifurcation is always super-critical and the bifurcating limit cycles are stable. The analysis is complemented with computations and some packet-level simulations as well. In terms of design, our study suggests that the presence of both forms of feedback may be detrimental to the performance of RCP.
We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. D time units later the timer expires, and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires, then the Lévy exponent does not change.Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.
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