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22
Heavy Traffic Limits for Queues with Many Deterministic Servers
"... Consider a sequence of stationary GI/D/N queues indexed by N with servers' utilization 1 #/ # N , # > 0. For such queues we show that the scaled waiting times NWN converge to the (finite) supremum of a Gaussian random walk with drift #. ..."
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Cited by 53 (5 self)
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Consider a sequence of stationary GI/D/N queues indexed by N with servers' utilization 1 #/ # N , # > 0. For such queues we show that the scaled waiting times NWN converge to the (finite) supremum of a Gaussian random walk with drift #.
Dynamic routing in largescale service systems with heterogeneous servers
, 2005
"... Motivated by modern call centers, we consider largescale service systems with multiple server pools and a single customer class. For such systems, we propose a simple routing rule which asymptotically minimizes the steadystate queue length and virtual waiting time. The proposed routing scheme is ..."
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Cited by 52 (12 self)
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Motivated by modern call centers, we consider largescale service systems with multiple server pools and a single customer class. For such systems, we propose a simple routing rule which asymptotically minimizes the steadystate queue length and virtual waiting time. The proposed routing scheme is FSF which assigns customers to the Fastest Servers First. The asymptotic regime considered is the HalfinWhitt manyserver heavytraffic regime, which we refer to as the Quality and Efficiency Driven (QED) regime; it achieves high levels of both service quality and system efficiency by carefully balancing between the two. Additionally, expressions are provided for system limiting performance measures based on diffusion approximations. Our analysis shows that in the QED regime this heterogeneous server system outperforms its homogeneous server counterpart.
Simulation run lengths to estimate blocking probabilities
 ACM Transactions on Modelling and Computer Simulation
, 1996
"... We derive formulas approximating the asymptotic variance of four estimators for the steadystate blocking probability in a multiserver loss system, exploiting diffusion process limits. These formulas can be used to predict simulation run lengths required to obtain desired statistical precision befor ..."
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Cited by 29 (19 self)
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We derive formulas approximating the asymptotic variance of four estimators for the steadystate blocking probability in a multiserver loss system, exploiting diffusion process limits. These formulas can be used to predict simulation run lengths required to obtain desired statistical precision before the simulation has been run, which can aid in the design of simulation experiments. They also indicate that one estimator can be much better than another, depending on the loading. An indirect estimator based on estimating the mean occupancy is significantly more (less) efficient than a direct estimator for heavy (light) loads. A major concern is the way computational effort scales with system size. For all the estimators, the asymptotic variance tends to be inversely proportional to the system size, so that the computational effort (regarded as proportional to the product of the asymptotic variance and the arrival rate) does not grow as system size increases. Indeed, holding the blocking probability fixed, the computational effort with a good estimator decreases to 0 as the system size increases. The asymptotic variance formulas also reveal the impact of the arrivalprocess and servicetime variability on the statistical precision. We validate these formulas by comparing them to exact numerical
Maximizing Cloud Providers Revenues via Energy Aware Allocation Policies
 in Proceedings of the 3rd IEEE International Conference on Cloud Computing (IEEE Cloud 2010
, 2010
"... Abstract—Cloud providers, like Amazon, offer their data centers ’ computational and storage capacities for lease to paying customers. High electricity consumption, associated with running a data center, not only reflects on its carbon footprint, but also increases the costs of running the data cente ..."
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Cited by 29 (3 self)
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Abstract—Cloud providers, like Amazon, offer their data centers ’ computational and storage capacities for lease to paying customers. High electricity consumption, associated with running a data center, not only reflects on its carbon footprint, but also increases the costs of running the data center itself. This paper addresses the problem of maximizing the revenues of Cloud providers by trimming down their electricity costs. As a solution allocation policies which are based on the dynamic powering servers on and off are introduced and evaluated. The policies aim at satisfying the conflicting goals of maximizing the users ’ experience while minimizing the amount of consumed electricity. The results of numerical experiments and simulations are described, showing that the proposed scheme performs well under different traffic conditions. I.
Service Level Differentiation in Call Centers with Fully Flexible Servers
, 2004
"... We study largescale service systems with multiple customer classes and many statistically identical servers. The following question is addressed: How many servers are required (staffing) and how does one match them with customers (control) in order to minimize staffing cost, subject to class level ..."
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Cited by 18 (8 self)
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We study largescale service systems with multiple customer classes and many statistically identical servers. The following question is addressed: How many servers are required (staffing) and how does one match them with customers (control) in order to minimize staffing cost, subject to class level quality of service constraints? We tackle this question by characterizing scheduling and staffing schemes that are asymptotically optimal in the limit, as system load grows to infinity. The asymptotic regimes considered are consistent with the Efficiency Driven (ED), Quality Driven (QD) and Quality and Efficiency Driven (QED) regimes, first introduced in the context of a single class service system. Our main findings are: a) Decoupling of staffing and control, namely (i) Staffing disregards the multiclass nature of the system and is analogous to the staffing of a single class system with the same aggregate demand and a single global quality of service constraint, and (ii) Class level service differentiation is obtained by using a simple Idle server based ThresholdPriority (ITP) control (with stateindependent thresholds), b) Robustness of the staffing and control rules: Our proposed SingleClass Staffing (SCS) rule and ITP control are approximately optimal
CALCULATING TRANSIENT CHARACTERISTICS OF THE ERLANG LOSS MODEL BY NUMERICAL TRANSFORM INVERSION
 Stochastic Models
"... In this paper we consider the classical Erlang loss model, i.e., the M/M/c/0 system with Poisson arrival process, exponential service times, c servers and no extra waiting space, where blocked calls are lost. We let the individual service rate be 1 and the arrival rate (which coincides with the offe ..."
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Cited by 17 (7 self)
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In this paper we consider the classical Erlang loss model, i.e., the M/M/c/0 system with Poisson arrival process, exponential service times, c servers and no extra waiting space, where blocked calls are lost. We let the individual service rate be 1 and the arrival rate (which coincides with the offered load) be a. We show how to compute several transient characteristics by numerical transform inversion. Transience arises by considering arbitrary fixed initial states.
The Impact of Dependent Service Times on LargeScale Service Systems
"... This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic ..."
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Cited by 17 (12 self)
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This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic limit, allowing dependence among the service times. That limit shows that the number of customers in the system at any time is approximately Gaussian, where the timevarying mean is unaffected by the dependence, but the timevarying variance is affected by the dependence. As a consequence, required staffing to meet customary qualityofservice targets in a largescale service system with finitely many servers based on a normal approximation is primarily affected by dependence among the service times through this timevarying variance. This paper develops formulas and algorithms to quantify the impact of the dependence among the service times upon that variance. The approximation applies directly to infiniteserver models, but also indirectly to associated finiteserver models, exploiting approximations based on the peakedness (the ratio of the variance to the mean in the infiniteserver model). Comparisons with simulations confirm that the approximations can be useful to assess the impact of the dependence.
Diffusion Approximations for a Single Node Accessed by CongestionControlled Sources
 IEEE Transactions on Automatic Control
, 1999
"... We consider simple models of congestion control in highspeed networks and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ONOFF type with exponential ON and OFF times, then, under a certain scaling, the steadystate distribution ..."
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Cited by 12 (5 self)
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We consider simple models of congestion control in highspeed networks and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ONOFF type with exponential ON and OFF times, then, under a certain scaling, the steadystate distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions. For the case where the source arrival process is Poisson and the service times are exponential, the steadystate distribution consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and servicetime distributions that are phasetype and show the impact of these distributions on the steadystate distribution of the number of sources in the system. We also establish an insensitivity to servicetime distributi...
Servicelevel differentiation in call centers with fully flexible servers
 Management Sci
"... doi 10.1287/mnsc.1070.0825 ..."
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Diffusion Approximations For Models of Congestion Control in HighSpeed Networks
 In Proc. IEEE Conf. Decision and Control
, 1998
"... We consider simple models of congestion control in highspeed networks and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ONOFF type with exponential ON and OFF times, then, under a certain scaling, the steadystate distribution ..."
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Cited by 6 (3 self)
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We consider simple models of congestion control in highspeed networks and develop diffusion approximations which could be useful for resource allocation. We first show that, if the sources are ONOFF type with exponential ON and OFF times, then, under a certain scaling, the steadystate distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions. For the case where the source arrival process is Poisson and the service times are exponential, the steadystate distribution consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and servicetime distributions that are phasetype and show the impact of these distributions on the steadystate distribution. We also establish an insensitivity to servicetime distribution when the arrival process is Poisson....