A fluid model for many-server queues with time-varying arrivals and phase-type service distribution

2012 ◽  
Vol 39 (4) ◽  
pp. 43-43
Author(s):  
Yunan Liu ◽  
Ward Whitt
Keyword(s):  
Fluid Model ◽  
Time Varying ◽  
Service Distribution ◽  
Phase Type ◽  
Many Server Queues

Fluid Limits for Multiclass Many-Server Queues with General Reneging Distributions and Head-of-the-Line Scheduling

2021 ◽  
Author(s):  
Amber L. Puha ◽  
Amy R. Ward
Keyword(s):  
Fluid Model ◽  
Time Varying ◽  
Fluid Limit ◽  
Control Policies ◽  
Model Solutions ◽  
Server Queue ◽  
Many Server Queues ◽  
Multiple Customer Classes ◽  
Limit Points

We describe a fluid model with time-varying input that approximates a multiclass many-server queue with general reneging distribution and multiple customer classes (specifically, the multiclass G/GI/N+GI queue). The system dynamics depend on the policy, which is a rule for determining when to serve a given customer class. The class of admissible control policies are those that are head-of-the-line (HL) and nonanticipating. For a sequence of many-server queues operating under admissible HL control policies and satisfying some mild asymptotic conditions, we establish a tightness result for the sequence of fluid scaled queue state descriptors and associated processes and show that limit points of such sequences are fluid model solutions almost surely. The tightness result together with the characterization of distributional limit points as fluid model solutions almost surely provides a foundation for the analysis of particular HL control policies of interest. We leverage these results to analyze a set of admissible HL control policies that we introduce, called weighted random buffer selection (WRBS), and an associated WRBS fluid model that allows multiple classes to be partially served in the fluid limit (which is in contrast to previously analyzed static priority policies).

scholarly journals Erlangian Approximations for Finite-Horizon Ruin Probabilities

2002 ◽  
Vol 32 (2) ◽  
pp. 267-281 ◽  
Author(s):  
Soren Asmussen ◽  
Florin Avram ◽  
Miguel Usabel
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Fluid Model ◽  
Ruin Probabilities ◽  
Approximation Procedure ◽  
Exponential Form ◽  
Probability Of Ruin ◽  
Phase Type ◽  
The Matrix ◽  
The Mean

AbstractFor the Cramér-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram & Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

scholarly journals Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Veena Goswami ◽  
M. L. Chaudhry
Keyword(s):  
Functional Equation ◽  
Busy Period ◽  
Stochastic Modelling ◽  
Lagrange Inversion ◽  
Special Cases ◽  
Service Distribution ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Laplace Transform ◽  
Number Of Customers

<p style='text-indent:20px;'>We give analytically explicit solutions for the distribution of the number of customers served during a busy period for the <inline-formula><tex-math id="M1">\begin{document}$ M^X/PH/1 $\end{document}</tex-math></inline-formula> queues when initiated with <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> customers. When customers arrive in batches, we present the functional equation for the Laplace transform of the number of customers served during a busy period. Applying the Lagrange inversion theorem, we provide a refined result to this functional equation. From a phase-type service distribution, we obtain the distribution of the number of customers served during a busy period for various special cases such as exponential, Erlang-k, generalized Erlang, hyperexponential, Coxian, and interrupted Poisson process. The results are exact, rapid and vigorous, owing to the clarity of the expressions. Moreover, we also consider computational results for several service-time distributions using our method. Phase-type distributions can approximate any non-negative valued distribution arbitrarily close, making them a useful practical stochastic modelling tool. These distributions have eloquent properties which make them beneficial in the computation of performance models.</p>

scholarly journals A TIME-VARYING CALL CENTER DESIGN VIA LAGRANGIAN MECHANICS

2009 ◽  
Vol 23 (2) ◽  
pp. 231-259 ◽  
Author(s):  
Robert C. Hampshire ◽  
Otis B. Jennings ◽  
William A. Massey
Keyword(s):  
Call Center ◽  
Fluid Model ◽  
Service Level Agreement ◽  
Service Level ◽  
Lagrangian Mechanics ◽  
Time Varying ◽  
Queuing Model ◽  
Service Network ◽  
Safety Rule ◽  
Loss Systems

We consider a multiserver delay queue with finite additional waiting spaces and time-varying arrival rates, where the customers waiting in the buffer may abandon. These are features that arise naturally from the study of service systems such as call centers. Moreover, we assume rewards for successful service completions and cost rates for service resources. Finally, we consider service-level agreements that constrain both the fractions of callers who abandon and the ones who are blocked.Applying the theory of Lagrangian mechanics to the fluid limit of a related Markovian service network model, we obtain near-profit-optimal staffing and provisioning schedules. The nature of this solution consists of three modes of operation. A key step in deriving this solution is combining the modified offered load approximation for loss systems with our fluid model. We use them to estimate effectively both our service-level agreement metrics and the profit for the original queuing model. Second-order profit improvements are achieved through a modified offered load version of the conventional square root safety rule.

scholarly journals PERFECT FLUID COSMOLOGICAL MODELS WITH TIME-VARYING CONSTANTS

2003 ◽  
Vol 12 (06) ◽  
pp. 1113-1129 ◽  
Author(s):  
JOSÉ ANTONIO BELINCHÓN ◽  
INDRAJIT CHAKRABARTY
Keyword(s):  
Cosmological Model ◽  
Perfect Fluid ◽  
Fluid Model ◽  
Time Varying ◽  
Bulk Viscous Fluid ◽  
Bulk Viscous ◽  
Varying Constants ◽  
Time Varying Constants ◽  
Limiting Case ◽  
Symmetry Method

In this paper, we study in detail a perfect fluid cosmological model with time-varying "constants" using dimensional analysis and the symmetry method. We examine the case of variable "constants" in detail without considering the perfect fluid model as a limiting case of a model with a causal bulk viscous fluid as discussed in a recent paper. We obtain some new solutions through the Lie method and show that when matter creation is considered, these solutions are physically relevant.

scholarly journals A NOTE ON MANY-SERVER FLUID MODELS WITH TIME-VARYING ARRIVALS

2018 ◽  
Vol 33 (3) ◽  
pp. 417-437 ◽  
Author(s):  
Zhenghua Long ◽  
Jiheng Zhang
Keyword(s):  
Service Time ◽  
Fluid Model ◽  
Convolution Equation ◽  
Fluid Models ◽  
Time Varying ◽  
One Dimensional ◽  
Interesting Connection ◽  
Service Times ◽  
Two Fluid

We extend the measure-valued fluid model, which tracks residuals of patience and service times, to allow for time-varying arrivals. The fluid model can be characterized by a one-dimensional convolution equation involving both the patience and service time distributions. We also make an interesting connection to the measure-valued fluid model tracking the elapsed waiting and service times. Our analysis shows that the two fluid models are actually characterized by the same one-dimensional convolution equation.

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