A Birth and Death Process Related to the Rogers-Ramanujan Continued Fraction

Abstract

Time-dependent system size probabilities of a birth and death process related to the Rogers-Ramanujan continued fraction are obtained. The range for the parameter in this continued fraction is obtained to ensure the positivity of the recursively defined birth and death rates. The general behavior of the birth and death rates is described and the asymptotic behavior of the transition probabilities and the (quasi)-stationary distribution is determined. For the transient case the birth and death process can be seen as a model in queuing theory where the length of the queue encourages customers to join the queue (the more the merrier). © 1998 Academic Press.