scholarly journals Perfect simulation of M/G/c queues

2015 ◽  
Vol 47 (04) ◽  
pp. 1039-1063
Author(s):  
Stephen B. Connor ◽  
Wilfrid S. Kendall
Keyword(s):  
Sample Path ◽  
Careful Analysis ◽  
Processor Sharing ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Random Assignment ◽  
Stable Case ◽  
Duration Distribution ◽  
Dynamic Reversibility ◽  
Service Duration

In this paper we describe a perfect simulation algorithm for the stable M/G/cqueue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/cqueue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service). The method exploited the fact that the workload process for the M/G/1 queue remains the same under different queueing disciplines, in particular under the processor sharing discipline, for which a dynamic reversibility property holds. We generalise Sigman's construction to the stable case by comparing the M/G/cqueue to a copy run under random assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled M/G/cqueues started respectively from the empty state and the state of the M/G/cqueue under random assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarise statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.

scholarly journals Perfect simulation of M/G/c queues

2015 ◽  
Vol 47 (4) ◽  
pp. 1039-1063 ◽  
Author(s):  
Stephen B. Connor ◽  
Wilfrid S. Kendall
Keyword(s):  
Sample Path ◽  
Careful Analysis ◽  
Processor Sharing ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Random Assignment ◽  
Stable Case ◽  
Duration Distribution ◽  
Dynamic Reversibility ◽  
Service Duration

In this paper we describe a perfect simulation algorithm for the stable M/G/c queue. Sigman (2011) showed how to build a dominated coupling-from-the-past algorithm for perfect simulation of the super-stable M/G/c queue operating under first-come-first-served discipline. Sigman's method used a dominating process provided by the corresponding M/G/1 queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service). The method exploited the fact that the workload process for the M/G/1 queue remains the same under different queueing disciplines, in particular under the processor sharing discipline, for which a dynamic reversibility property holds. We generalise Sigman's construction to the stable case by comparing the M/G/c queue to a copy run under random assignment. This allows us to produce a naïve perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled M/G/c queues started respectively from the empty state and the state of the M/G/c queue under random assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarise statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.

scholarly journals Chains with unbounded variable length memory: perfect simulation and a visible regeneration scheme

2011 ◽  
Vol 43 (03) ◽  
pp. 735-759 ◽  
Author(s):  
Sandro Gallo
Keyword(s):  
Transition Probabilities ◽  
Natural Extension ◽  
Continuity Condition ◽  
Variable Length ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Deterministic Function ◽  
The Family ◽  
Context Tree ◽  
Probabilistic Context

We present a new perfect simulation algorithm for stationary chains having unbounded variable length memory. This is the class of infinite memory chains for which the family of transition probabilities is represented by a probabilistic context tree. We do not assume any continuity condition: our condition is expressed in terms of the structure of the context tree. More precisely, the length of the contexts is a deterministic function of the distance to the last occurrence of some determined string of symbols. It turns out that the resulting class of chains can be seen as a natural extension of the class of chains having a renewal string. In particular, our chains exhibit a visible regeneration scheme.

scholarly journals Revisiting Feller Diffusion: Derivation and Simulation

2019 ◽  
pp. 1-15
Author(s):  
Ranjiva Munasinghe ◽  
Leslie Kanthan ◽  
Pathum Kossinna
Keyword(s):  
Fourier Transform ◽  
Probability Density Function ◽  
Probability Density ◽  
Method Of Characteristics ◽  
Sample Path ◽  
Planck Equation ◽  
Simulation Algorithm ◽  
Quantitative Finance ◽  
Potential Applications ◽  
The Fourier Transform

We propose a simpler derivation of the probability density function of Feller Diffusion by using the Fourier Transform on the associated Fokker-Planck equation and then solving the resulting equation via the Method of Characteristics. We use the derived probability density to formulate an exact simulation algorithm whereby a sample path increment is drawn directly from the density. We then proceed to use the simulation to verify key statistical properties of the process such as the moments and the martingale property. The simulation is also used to confirm properties related to hitting time probabilities. We also mention potential applications of the simulation in the setting of quantitative finance.

scholarly journals Backward Coalescence Times for Perfect Simulation of Chains with Infinite Memory

2012 ◽  
Vol 49 (02) ◽  
pp. 319-337 ◽  
Author(s):  
Emilio De Santis ◽  
Mauro Piccioni
Keyword(s):  
Stationary Process ◽  
Past History ◽  
A Priori ◽  
Sufficient Conditions ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Infinite Memory ◽  
Countable State ◽  
Weaker Conditions ◽  
Coalescence Times

This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernández and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We define backward coalescence times for these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, Fernández and Ferrari (2002). We discuss how to construct backward coalescence times (i) by means of information depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitable coalescing events.

scholarly journals Perfect simulation of Hawkes processes

2005 ◽  
Vol 37 (03) ◽  
pp. 629-646 ◽  
Author(s):  
Jesper Møller ◽  
Jakob G. Rasmussen
Keyword(s):  
Distribution Function ◽  
Edge Effects ◽  
Conditional Independence ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Hawkes Processes ◽  
Empirical Results ◽  
Poisson Cluster ◽  
Cluster Processes

Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented.

scholarly journals Chains with unbounded variable length memory: perfect simulation and a visible regeneration scheme

2011 ◽  
Vol 43 (3) ◽  
pp. 735-759 ◽  
Author(s):  
Sandro Gallo
Keyword(s):  
Transition Probabilities ◽  
Natural Extension ◽  
Continuity Condition ◽  
Variable Length ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Deterministic Function ◽  
The Family ◽  
Context Tree ◽  
Probabilistic Context

We present a new perfect simulation algorithm for stationary chains having unbounded variable length memory. This is the class of infinite memory chains for which the family of transition probabilities is represented by a probabilistic context tree. We do not assume any continuity condition: our condition is expressed in terms of the structure of the context tree. More precisely, the length of the contexts is a deterministic function of the distance to the last occurrence of some determined string of symbols. It turns out that the resulting class of chains can be seen as a natural extension of the class of chains having a renewal string. In particular, our chains exhibit a visible regeneration scheme.

scholarly journals Backward Coalescence Times for Perfect Simulation of Chains with Infinite Memory

2012 ◽  
Vol 49 (2) ◽  
pp. 319-337 ◽  
Author(s):  
Emilio De Santis ◽  
Mauro Piccioni
Keyword(s):  
Stationary Process ◽  
Past History ◽  
A Priori ◽  
Sufficient Conditions ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Infinite Memory ◽  
Countable State ◽  
Weaker Conditions ◽  
Coalescence Times

This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernández and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We definebackward coalescence timesfor these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, Fernández and Ferrari (2002). We discuss how to construct backward coalescence times (i) by means ofinformation depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitablecoalescing events.

scholarly journals Patterns in Student Assignment to Elementary School Classrooms

2013 ◽  
Vol 21 ◽  
pp. 51
Author(s):  
Ryan Bosworth ◽  
Hao Li
Keyword(s):  
North Carolina ◽  
Elementary School ◽  
Elementary Schools ◽  
Elementary School Students ◽  
Value Added ◽  
Careful Analysis ◽  
Random Assignment ◽  
School Students ◽  
5Th Grade ◽  
Classroom Assignment

In an effort to better understand aggregate patterns in the way elementary school students are assigned to classes, we conduct a careful analysis of observed classroom assignment outcomes in the 5th grade in North Carolina elementary schools. First, we model the probability that a pair of students are classmates as a function of the characteristics of that pair of students. This novel methodological technique enables us to directly observe the degree to which actual assignment patterns differ from what might be expected under random assignment for a wide variety of student characteristics. Second, we analyze patterns in classroom assignment and discuss the implications of these patterns. We show that classroom assignments tend to deviate from random assignment in a way that tends to group similar students and that these deviations tend to be greatly magnified in Magnet schools. Importantly, we find evidence that administrators sort students based on attributes not normally observable by researchers. These findings have important implications for researchers using value-added modeling (VAM) techniques. Finally, we find that classroom assignment patterns are generally stable across the racial, income, and geographic characteristics of schools.

scholarly journals Perfect simulation of Hawkes processes

2005 ◽  
Vol 37 (3) ◽  
pp. 629-646 ◽  
Author(s):  
Jesper Møller ◽  
Jakob G. Rasmussen
Keyword(s):  
Distribution Function ◽  
Edge Effects ◽  
Conditional Independence ◽  
Perfect Simulation ◽  
Simulation Algorithm ◽  
Hawkes Processes ◽  
Empirical Results ◽  
Poisson Cluster ◽  
Cluster Processes

Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented.

Sign in / Sign up

Export Citation Format

Share Document