Idle Servers Research Articles

Performance evaluation of two Markovian retrial queueing model with balking and feedback

Abstract In this paper, we consider the performance evaluation of two retrial queueing system. Customers arrive to the system, if upon arrival, the queue is full, the new arriving customers either move into one of the orbits, from which they make a new attempts to reach the primary queue, until they find the server idle or balk and leave the system, these later, and after getting a service may comeback to the system requiring another service. So, we derive for this system, the joint distribution of the server state and retrial queue lengths. Then, we give some numerical results that clarify the relationship between the retrials, arrivals, balking rates, and the retrial queue length.

Optimal Control of a Server Farm

We consider a server farm consisting of ample exponential servers, that serve a Poisson stream of arriving customers. Each server can be either busy, idle or off. An arriving customer will immediately occupy an idle server, if there is one, and otherwise, an off server will be turned on and start servicing this customer. Upon service completion, it can be decided to keep the server idle or to turn it off. It costs per unit time to keep servers idle, and it costs to turn an off server on. We derive structural properties of the dynamic control policy minimizing the expected total discounted cost. The structural results are extended to the average cost criterion. We also propose and analyze a simple control policy, and numerically investigate its performance.

Multi-server machine repair problems under a (V, R) synchronous single vacation policy

This paper considers a machine repair problem with M operating machines and S standbys, in which R repairmen are responsible for supervising these machines and operate a (V,R) vacation policy. With such policy, if the number of the failed machines is reduced to R−V (R>V) (there exists V idle repairmen) at a service completion, these V idle servers will together take a synchronous vacation (or leave for other secondary job). Upon returning from the vacation, they do not take a vacation again and remain idle until the first arriving failed machine arrives. The steady-state probabilities are solved in terms of matrix forms and the system performance measures are obtained. Algorithmic procedures are provided to deal with the optimization problem of discrete/continuous decision variables while maintaining a minimum specified level of system availability.

An Infinite Server System with General Packing Constraints

We consider a service system model primarily motivated by the problem of efficient assignment of virtual machines to physical host machines in a network cloud, so that the number of occupied hosts is minimized. There are multiple input flows of different type customers, with a customer mean service time depending on its type. There is an infinite number of servers. A server-packing configuration is the vector k = {ki}, where ki is the number of type i customers the server “contains.” Packing constraints must be observed; namely, there is a fixed finite set of configurations k that are allowed. Service times of different customers are independent; after a service completion, each customer leaves its server and the system. Each new arriving customer is placed for service immediately; it can be placed into a server already serving other customers (as long as packing constraints are not violated), or into an idle server. We consider a simple parsimonious real-time algorithm, called Greedy, that attempts to minimize the increment of the objective function [Formula: see text], α > 0, caused by each new assignment; here Xk is the number of servers in configuration k. (When α is small, [Formula: see text] approximates the total number [Formula: see text] of occupied servers.) Our main results show that certain versions of the Greedy algorithm are asymptotically optimal, in the sense of minimizing [Formula: see text] in stationary regime as the input flow rates grow to infinity. We also show that in the special case when the set of allowed configurations is determined by vector-packing constraints, the Greedy algorithm can work with aggregate configurations as opposed to exact configurations k, thus reducing computational complexity while preserving the asymptotic optimality.

Cluster Energy Optimization: An Algorithmic Approach

In fact, Gartner projected global revenue for cloud computing to reach almost $150 billion by 2014. However, The 2011 market is already approx $68 billion globally. With increase in web technologies and Internet, a proportional increase in Cloud computing technologies has been cited. Cloud computing has been emerging as a flexible and powerful computational architecture to offer ubiquitous services to users. A variety of hardware and software resources are integrated together as a resource pool, the software is no longer resided in a single hardware environment, it is performed upon the schedule of the resource pool for optimized resource utilization. The optimization of energy consumption in the cloud computing environment is the question how to use various energy conservation strategies to efficiently allocate resources. The need of different resources in cloud environment is unpredictable. It is observed that load management in cloud is utmost needed in order to provide QOS. The jobs at over-loaded physical machine are shifted to under-loaded physical machine and turning the idle machine off in order to provide green cloud. For energy optimization, DVFS and Power-Nap are good strategies. As much of this energy is wasted in idle systems: in typical deployments, server utilization is below 30%, but idle servers still consume 60% of their peak power draw. In this paper, we have proposed an algorithm for energy optimization having the constraint QOS and SLA.

Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward

The M/M/k/setup model, where there is a penalty for turning servers on, is common in data centers, call centers and manufacturing systems. Setup costs take the form of a time delay, and sometimes there is additionally a power penalty, as in the case of data centers. While the M/M/1/setup was exactly analyzed in 1964, no exact analysis exists to date for the M/M/k/setup with k>1. In this paper we provide the first exact, closed-form analysis for the M/M/k/setup and some of its important variants including systems in which idle servers delay for a period of time before turning off or can be put to sleep. Our analysis is made possible by our development of a new technique, Recursive Renewal Reward (RRR), for solving Markov chains with a repeating structure. RRR uses ideas from renewal reward theory and busy period analysis to obtain closed-form expressions for metrics of interest such as the transform of time in system and the transform of power consumed by the system. The simplicity, intuitiveness, and versatility of RRR makes it useful for analyzing Markov chains far beyond the M/M/k/setup. In general, RRR should be used to reduce the analysis of any 2-dimensional Markov chain which is infinite in at most one dimension and repeating to the problem of solving a system of polynomial equations. In the case where all transitions in the repeating portion of the Markov chain are skip-free and all up/down arrows are unidirectional, the resulting system of equations will yield a closed-form solution.

Simple and Effective Dynamic Provisioning for Power-Proportional Data Centers

Energy consumption represents a significant cost in data center operation. A large fraction of the energy, however, is used to power idle servers when the workload is low. Dynamic provisioning techniques aim at saving this portion of the energy, by turning off unnecessary servers. In this paper, we explore how much gain knowing future workload information can bring to dynamic provisioning. In particular, we develop online dynamic provisioning solutions with and without future workload information available. We first reveal an elegant structure of the offline dynamic provisioning problem, which allows us to characterize the optimal solution in a “divide-andconquer” manner. We then exploit this insight to design two online algorithms with competitive ratios 2 - α and e/(e - 1 + α), respectively, where 0 ≤ α ≤ 1 is the normalized size of a look-ahead window in which future workload information is available. A fundamental observation is that future workload information beyond the full-size look-ahead window (corresponding to α = 1) will not improve dynamic provisioning performance. Our algorithms are decentralized and easy to implement. We demonstrate their effectiveness in simulations using real-world traces.

Analysis of the Machine Repair Models With Multi-Threshold Synchronous Vacations

Abstract In this paper, we present an analysis of a machine repair problem with M operating machines and S standbys, in which c repairmen are responsible for supervising these machines and follow a multi-threshold, synchronous vacation policy. With such a policy, at a repair completion instant, if the number of the failed machines in the system is less than a preset threshold value, partial idle servers together take a single vacation (or leave for a random amount of time doing other secondary job). At the end of a vacation, they return to repair the failed machines (or waits for the arrival). The steady-state probabilities are solved by using the matrix-analytic method, and formulae of system performance measures are thereby obtained. A cost model is then developed to formulate an optimization problem to find the minimum cost. The direct search and quasi-Newton methods are implemented to determine the optimal number of servers, the optimal threshold policy and the optimal repair rate.

Strategic joining in M/M/1 retrial queues

The equilibrium and socially optimal balking strategies are investigated for unobservable and observable single-server classical retrial queues. There is no waiting space in front of the server. If an arriving customer finds the server idle, he occupies the server immediately and leaves the system after service. Otherwise, if the server is found busy, the customer decides whether or not to enter a retrial pool with infinite capacity and becomes a repeated customer, based on observation of the system and the reward–cost structure imposed on the system. Accordingly, two cases with respect to different levels of information are studied and the corresponding Nash equilibrium and social optimization balking strategies for all customers are derived. Finally, we compare the equilibrium and optimal behavior regarding these two information levels through numerical examples.

Formula omitted] with staggered setup

We consider the M/G/k/staggered-setup, where idle servers are turned off to save cost, necessitating a setup time for turning a server back on; however, at most one server may be in setup mode at any time. We show that, for exponentially distributed setup times, the response time of an M/G/k/staggered-setup approximately decomposes into the sum of the response time for an M/G/k and the setup time, where the approximation is nearly exact. This generalizes a prior decomposition result for an M/M/k/staggered-setup.

Stochastic service systems, random interval graphs and search algorithms

ABSTRACTWe consider several stochastic service systems, and study the asymptotic behavior of the moments of various quantities that have application to models for random interval graphs and algorithms for searching for an idle server or for an vacant or occupied waiting station. In some cases the moments turn out to involve Lambert series for the generating functions for the sums of powers of divisors of positive integers. For these cases we are able to obtain complete asymptotic expansions for the moments of the quantities in question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 421–442, 2014

A multi-server queueing system with service interruption, partial protection and repetition of service

A queueing model consisting of two multi-server service systems is considered. Primary customers arrive at a multi-server queueing system-1 having an infinite buffer. The input flow is described by a MAP (Markovian Arrival Process). The service time of a primary customer has a PH (Phase-type) distribution. Besides the primary customers, a MAP of interruptions arrives to the system. An interruption removes one of the primary customers from the service if the state (phase) of its PH service process does not belong to some given set of so called protected phases when an interruption is successful. The interrupted customer leaves the system permanently with some probability. With complementary probability, the interrupted primary customer moves for service to system-2. This system consists of K independent identical servers and has no buffer. If all K servers are busy at the moment of a primary customer interruption, this customer will be lost. Otherwise, this primary customer starts the service in an arbitrary idle server of system-2. It is assumed that the service time of a primary customer by a server of system-2 has a PH distribution. Upon completion of the service at system-2, the customer becomes priority customer. If, at the service completion moment, there are free servers at system-1, the priority customer immediately starts getting the service at system-1. It is assumed that the service time of a priority customer by a server of system-1 has a PH distribution and this service can not be interrupted. If, at the moment when the primary customer finished the service at system-2, there are no idle servers at system-1, this customer is placed into the finite buffer for priority customers of capacity K. The customers will be picked up for the service according to the FIFO discipline. When a server of system-1 becomes free, it takes for service a priority customer from this buffer, if any. Type-1 customers are picked-up from the infinite buffer only if the buffer for priority customers is empty at the service completion moment at system-1. Behavior of this system is described by a multi-dimensional Markov chain. Algorithms for checking ergodicity condition and computing the stationary distribution are presented. Formulas for computing important performance measures of the system are derived.

On Fair Routing from Emergency Departments to Hospital Wards: QED Queues with Heterogeneous Servers

The interface between an emergency department and internal wards is often a hospital's bottleneck. Motivated by this interaction in an anonymous hospital, we analyze queueing systems with heterogeneous server pools, where the pools represent the wards, and the servers are beds. Our queueing system, with a single centralized queue and several server pools, forms an inverted-V model. We introduce the randomized most-idle (RMI) routing policy and analyze it in the quality- and efficiency-driven regime, which is natural in our setting. The RMI policy results in the same server fairness (measured by idleness ratios) as the longest-idle-server-first (LISF) policy, which is commonly used in call centers and considered fair. However, the RMI policy utilizes only the information on the number of idle servers in different pools, whereas the LISF policy requires information that is unavailable in hospitals on a real-time basis. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

TACOMA

A two-tier Internet data center management scheme, TACOMA, with thermal-aware server provisioning (TASP) in one tier, and thermal-aware workload distribution (TAWD) in the other is proposed. TASP and TAWD coordinate to maximize the energy savings by leveraging the workload dynamics, at coarse and fine time scale, respectively. TACOMA is aware of the QoS constraints, the energy proportionality of servers, and the potential trade-off between cooling and computing power. The obtained energy savings are a combination of suspending idle servers, using servers at their peak efficiency, and avoiding heat recirculation.

Multiclass multiserver queueing system in the Halfin–Whitt heavy traffic regime: asymptotics of the stationary distribution

We consider a heterogeneous queueing system consisting of one large pool of O(r) identical servers, where r→∞ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfin–Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where a>0 is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue Q r (∞). While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both Q r (∞) and the number of idle servers. The bounds are uniform w.r.t. parameter r and the service policy. In particular, we show that $\limsup_{r} \mathbb {E}\exp(\theta r^{-{1\over2}}Q^{r}(\infty))<\infty$ . Therefore, the sequence $r^{-{1\over2}}Q^{r}(\infty)$ is tight and has a uniform exponential tail bound. We further consider the system with strictly positive abandonment rates, and show that in this case every weak limit $\hat{Q}(\infty)$ of $r^{-{1\over2}}Q^{r}(\infty)$ has a sub-Gaussian tail. Namely, $\mathbb {E}[\exp(\theta(\hat{Q}(\infty ))^{2})]<\infty$ , for some θ>0.

Dynamic routing in distinguishable parallel queues: an application of product returns for remanufacturing

This paper deals with the dynamic routing of product returns in distinguishable parallel queues. Several vendors alongside an original equipment manufacturer are available in provision of remanufacturing service. Each has its own queue. The stream of the product returns follow a stochastic process. A central controller is employed to decide to which vendor an incoming product is sent to avoid excessive queues in front of some vendors and idle servers in the others. We develop models and index-based heuristics to support the dynamic routing decisions so as to minimize the overall recovering costs. The product concerned exemplifies a short-life cycle due to, for example, technology advance. Long delay during the remanufacturing process will render a substantial deterioration of reselling prices. Hence, in the paper we contend that the cost incurred for remanufacturing a product should take explicit account of the impact of long delays in the lead time. Both theoretical and simulation studies demonstrate the effectiveness of the Restless Bandit approach deployed to the dynamic routing of product returns among multiple vendors.

ECO–ALOC: Energy-efficient resource allocation for cluster-based software routers

In this paper, we explore the use of cluster-based software routers as a way to share hardware resources, reduce costs, and save energy. We propose ECO–ALOC, an energy-efficient resource allocation mechanism that reconfigures the router according to the traffic demands using two modules. The first module provides fine-grained energy consumption control by switching CPU operation frequencies among three special frequencies. The second module provides long-term power savings by using virtual router migration to consolidate the load and shut idle servers down. We simulate a cluster-based software router using real traffic traces to evaluate our proposal. Results show that ECO–ALOC provides power savings of up to 93% depending on the cluster load scenario.

A product form solution to a system with multi-type jobs and multi-type servers

We consider a memoryless single station service system with servers \(\mathcal{S}=\{m_{1},\ldots,m_{K}\}\), and with job types \(\mathcal{C}=\{a,b,\ldots\}\). Service is skill-based, so that server m i can serve a subset of job types \(\mathcal{C}(m_{i})\). Waiting jobs are served on a first-come-first-served basis, while arriving jobs that find several idle servers are assigned to a feasible server randomly. We show that there exist assignment probabilities under which the system has a product-form stationary distribution, and obtain explicit expressions for it. We also derive waiting time distributions in steady state.

Analysis of discrete-time MAP/G/1 queue under workload control

In this paper, we analyze the discrete-time MAP/G/1 queue under the D-policy in which the idle server resumes its service only when the accumulated workload exceeds the predetermined threshold D. We first derive the probability generating functions of the queue length, workload, waiting time, and sojourn time distributions. Then we derive the mean performance measures. Lastly we present our numerical experience. Through the numerical examples we show the effects of the threshold and the variability of the service times on the mean performance measures, and the effect of the correlation between arrivals.

A Markov‐modulated fluid flow queueing model under D‐policy

SUMMARYWe consider a Markov‐modulated fluid flow queueing model under D‐policy. As soon as the fluid level reaches zero, the server becomes idle. During the idle period, fluid arrives from outside according to an underlying continuous time Markov chain (UMC) and the idle server does not process the fluid. We consider two increase patterns of fluid during the idle period: vertical increase (Type‐V) and linear increase (Type‐L). The idle server is reactivated only when the cumulative fluid level in the system exceeds a predetermined threshold value D. We derive the distributions of fluid level and mean performance measures for both types. We also present cost optimization model to minimize average operating cost per unit time. Copyright © 2011 John Wiley &amp; Sons, Ltd.