Compound Binomial Model with Batch Markovian Arrival Process

A compound binomial model with batch Markovian arrival process was studied, and the specific definitions are introduced. We discussed the problem of ruin probabilities. Specially, the recursion formulas of the conditional finite-time ruin probability are obtained and the numerical algorithm of the conditional finite-time nonruin probability is proposed. We also discuss research on the compound binomial model with batch Markovian arrival process and threshold dividend. Recursion formulas of the Gerber–Shiu function and the first discounted dividend value are provided, and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations were presented.

A Queueing Model and Analysis for Autonomous Vehicles on Highways

We investigate the effects of autonomous vehicles (AVs) on highway congestion. AVs have the potential to significantly reduce highway congestion because they can maintain smaller intervehicle gaps and travel together in larger platoons than human-driven vehicles (HVs). Various policies have been proposed to regulate AV travel on highways, yet no in-depth comparison of these policies exists. To address this shortcoming, we develop a queueing model for a multilane highway and analyze two policies: the designated-lane policy (“D policy”), under which one lane is designated to AVs, and the integrated policy (“I policy”), under which AVs travel together with HVs in all lanes. We connect the service rate to intervehicle gaps (governed by a Markovian arrival process) and congestion, and measure the performance using mean travel time and throughput. Our analysis shows that although the I policy performs at least as well as a benchmark case with no AVs, the D policy outperforms the benchmark only when the highway is heavily congested and AVs constitute the majority of vehicles; in such a case, this policy may outperform the I policy only in terms of throughput. These findings caution against recent industry and government proposals that the D policy should be employed at the beginning of the mass appearance of AVs. Finally, we calibrate our model to data and show that for highly congested highways, a moderate number of AVs can make a substantial improvement (e.g., 22% AVs can improve throughput by 30%), and when all vehicles are AVs, throughput can be increased by over 400%. This paper was accepted by Jayashankar Swaminathan, operations management.

Analysis of retrial queue with heterogeneous servers and Markovian arrival process

Multi-server retrial queueing system with heterogeneous servers is analyzed. Requests arrive to the system according to the Markovian arrival process. Arriving primary requests and requests retrying from orbit occupy an available server with the highest service rate, if there is any available server. Otherwise, the requests move to the orbit having an infinite capacity. The total retrial rate infinitely increases when the number of requests in orbit increases. Service periods have exponential distribution. Behavior of the system is described by multi-dimensional continuous-time Markov chain which belongs to the class of asymptotically quasi-toeplitz Markov chains. This allows to derive simple and transparent ergodicity condition and compute the stationary probabilities distribution of chain states. Presented numerical results illustrate the dynamics of some system effectiveness indicators and the importance of considering of correlation in the requests arrival process.