Alternating renewal processes with instantaneous rewards

Abstract

Consider an alternating renewal process that, over time, alternates between two states (up and down), starting in upstate at time 0. Associated with each up interval a reward which is a function of the interval length. Similarly, we associate with each down interval a reward which depends on the length of it through some function. We call the total reward earned in the time interval $[0,t],t\ge 0,$ an instantaneous alternating renewal reward process. In this article, we derive the probability distribution of the total reward and its expected value. The results are presented in the form of Laplace transforms.

Keywords:

• Laplace transform
• Poisson process
• renewal process

2010 MSC:

• 60K05
• 62G55

Additional information

Funding

This work was supported by the Directorate of Research and Community Service, Ministry of Research, Technology, and Higher Education of the Republic of Indonesia [grant number 47/SP2H/DRPM/LPPM-UNJ/II/2018].