The departure process for the M/M/1 queue

1986 ◽  
Vol 23 (1) ◽  
pp. 249-255 ◽  
Author(s):  
John R. Hubbard ◽  
Claude Dennis Pegden ◽  
Matthew Rosenshine
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Arrival Rate ◽  
Form Solution ◽  
Service Rate ◽  
Time Interval ◽  
Simple Computation ◽  
Departure Process ◽  
Computational Representations

The problem of determining the probability of j departures during the time interval (0, t) from an M/M/l queue empty at t = 0 is considered. A closed-form solution is obtained. It is shown that this solution is unique and invariant under interchanging the arrival rate and service rate. Finally, sample computational representations of the solution are developed and results of a simple computation are provided.

The departure process for the M/M/1 queue

1986 ◽  
Vol 23 (01) ◽  
pp. 249-255 ◽  
Author(s):  
John R. Hubbard ◽  
Claude Dennis Pegden ◽  
Matthew Rosenshine
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Arrival Rate ◽  
Form Solution ◽  
Service Rate ◽  
Time Interval ◽  
Simple Computation ◽  
Departure Process ◽  
Computational Representations

The problem of determining the probability of j departures during the time interval (0, t) from an M/M/l queue empty at t = 0 is considered. A closed-form solution is obtained. It is shown that this solution is unique and invariant under interchanging the arrival rate and service rate. Finally, sample computational representations of the solution are developed and results of a simple computation are provided.

Real Time Trajectory Modification Using a Closed Form Solution

1992 ◽  
Author(s):  
Michael R. Hummels ◽  
Raymond J. Cipra
Keyword(s):  
Closed Form ◽  
Real Time ◽  
Closed Form Solution ◽  
Form Solution ◽  
Time Interval ◽  
Efficient Manner ◽  
Planning Strategy ◽  
Higher Derivatives ◽  
On Line ◽  
Line Trajectory

Abstract An on-line trajectory modification and path planning strategy is developed which will allow a robot to respond in an efficient manner to real time sensory input. The approach developed here eliminates the need for solving many equations by developing a closed form algorithm. It uses two fourth order curves for the transition phases with a constant velocity section in between. Although this is done by providing additional constraints to the curve, it makes the problem of determining the trajectory much easier to solve, while providing continuous higher derivatives. It also provides a safe and efficient way of modifying trajectories based on the robots joint rate limits, joint acceleration limits, jerk limits, and desired time interval between trajectory modifications for a 4-1-4 trajectory. This method involves the solution of one second order equation and is directed toward real time applications.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

2021 ◽  
pp. 1-10
Author(s):  
Puneet Pasricha ◽  
Anubha Goel
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stochastic Volatility Model ◽  
Strike Price ◽  
Volatility Model ◽  
Exchange Options ◽  
Pricing Formula ◽  
Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

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