Optimal Open-Loop Routing and Threshold-Based Allocation in TWO Parallel QUEUEING Systems with Heterogeneous Servers

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties of the systems operating under the introduced policies and their performance characteristics.

Algorithmic Analysis of Finite-Source Multi-Server Heterogeneous Queueing Systems

The paper deals with a finite-source queueing system serving one class of customers and consisting of heterogeneous servers with unequal service intensities and of one common queue. The main model has a non-preemptive service when the customer can not change the server during its service time. The optimal allocation problem is formulated as a Markov-decision one. We show numerically that the optimal policy which minimizes the long-run average number of customers in the system has a threshold structure. We derive the matrix expressions for performance measures of the system and compare the main model with alternative simplified queuing systems which are analysed for the arbitrary number of servers. We observe that the preemptive heterogeneous model operating under a threshold policy is a good approximation for the main model by calculating the mean number of customers in the system. Moreover, using the preemptive and non-preemptive queueing models with the faster server first policy the lower and upper bounds are calculated for this mean value.

Transient Analysis of Power Management in Wireless Sensor Network With Start-Up Times and Threshold Policy

Queueing models play a significant role in analysing the performance of power management systems in various electronic devices and communication systems. This paper adopts a multiple vacation queueing model with a threshold policy to analyse the power-saving mechanisms of the wireless sensor network(WSN) using the Dynamic Power Management technique. The proposed system consists of a busy state(transmit state), wake-up state, shutdown state and inactive state. In this model, the server switches to a shutdown state for a random duration of time after serving all the events(data packets) in the busy state. Events that arrive during the shutdown period cannot be served until the system size reaches the predetermined threshold value of k and further it requires start-up time and a change of state to resume service. At the end of the shutdown period, if the system size is less than k, then the server begins the inactive period; otherwise, the server switches to the wake-up state. For this system, an explicit expression for the transient and steady-state solution is computed in a closed-form. Furthermore, performance indices such as mean, variance, probability that the server is in various stages of power management modes and mean power consumption are computed. Finally, graphical illustrations are made to understand the effect of the parameters on the performance of the system.

Simulation-Based Study of Biological Systems with Threshold Policy by a Differential Linear Complementarity System

Threshold policy is more realistic than continuous control for biological system management. Most related works are devoted to studying a single-threshold value for one single population, thereby avoiding complicated mathematical analysis of the nonsmooth differential equations. Based on the fact that numerical simulations play an important role in analyzing and understanding the intrinsic mechanism of a biological experiment and system, we hereby propose a differential linear complementarity system to reformulate the biological system with threshold policy. Using this method, we can transform a biological system with multiple-threshold values for one or more population to a differential linear complementarity system, where the corresponding dynamics can be investigated numerically by various algorithms for the complementarity problem. Firstly, the well-posedness of solutions of the differential linear complementarity system and its discretized method are derived explicitly. Then we illustrate the application of our approach to two systems which are a population harvesting system with threshold policy and an HIV replication system with threshold therapy, respectively. Numerical results demonstrate that those nonsmooth biological systems exhibit much more complex dynamics than the corresponding smooth systems. These results also validate the effectiveness and simplicity of the method that reformulates a common biological system with multiple-threshold policy by a differential linear complementarity system.

Estimation of the Optimal Threshold Policy in a Queue with Heterogeneous Servers Using a Heuristic Solution and Artificial Neural Networks

This paper deals with heterogeneous queues where servers differ not only in service rates but also in operating costs. The classical optimisation problem in queueing systems with heterogeneous servers consists in the optimal allocation of customers between the servers with the aim to minimise the long-run average costs of the system per unit of time. As it is known, under some assumptions the optimal allocation policy for this system is of threshold type, i.e., the policy depends on the queue length and the state of faster servers. The optimal thresholds can be calculated using a Markov decision process by implementing the policy-iteration algorithm. This algorithm may have certain limitations on obtaining a result for the entire range of system parameter values. However, the available data sets for evaluated optimal threshold levels and values of system parameters can be used to provide estimations for optimal thresholds through artificial neural networks. The obtained results are accompanied by a simple heuristic solution. Numerical examples illustrate the quality of estimations.

On the Cluster Admission Problem for Cloud Computing

Cloud computing providers face the problem of matching heterogeneous customer workloads to resources that will serve them. This is particularly challenging if customers, who are already running a job on a cluster, scale their resource usage up and down over time. The provider therefore has to continuously decide whether she can add additional workloads to a given cluster or if doing so would impact existing workloads’ ability to scale. Currently, this is often done using simple threshold policies to reserve large parts of each cluster, which leads to low efficiency (i.e., low average utilization of the cluster). We propose more sophisticated policies for controlling admission to a cluster and demonstrate that they significantly increase cluster utilization. We first introduce the cluster admission problem and formalize it as a constrained Partially Observable Markov Decision Process (POMDP). As it is infeasible to solve the POMDP optimally, we then systematically design admission policies that estimate moments of each workload’s distribution of future resource usage. Via extensive simulations grounded in a trace from Microsoft Azure, we show that our admission policies lead to a substantial improvement over the simple threshold policy. We then show that substantial further gains are possible if high-quality information is available about arriving workloads. Based on this, we propose an information elicitation approach to incentivize users to provide this information and simulate its effects.

Rich dynamics of a Filippov avian-only influenza model with a nonsmooth separation line

Depopulation of birds has been authenticated to be an effective measure in controlling avian influenza transmission. In this work, we establish a Filippov avian-only model incorporating a threshold policy control. We choose the index—the maximum between the infected threshold level $I_{T}$ I T and the product of the number of susceptible birds S and a ratio threshold value ξ—to decide on whether to trigger the control measures or not, which then leads to a discontinuous separation line and two pieces of sliding-mode domains. Meanwhile, one more sliding-mode domain gives birth to more complex dynamics. We investigate the global dynamical behavior of the Filippov model, including the real and/or virtual equilibria and the two sliding modes and their dynamics. The solutions will eventually stabilize at the real endemic equilibrium of the subsystem or the pseudoequilibria on the two sliding modes due to different threshold values. Therefore an effective and efficient threshold policy is essential to control the influenza by driving the number of infected birds below a certain level or at a previously given level.