The convexity of loss rate in an Erland loss system and sojourn in an Erlang delay system with respect to arrival and service rates

The Erlang loss system is a system in which customers arriving in a Poisson stream of arrival rate λ are served by a group of n servers, with a mean service time of 1/μ. Arrivals that find all servers busy are blocked and cleared from the system. The Erlang delay system is a system in which customers who find all servers busy wait in a queue until served by the first available server. Proofs of joint-convexity results in the Erlang delay system and Erlang loss system are presented, using induction on the number of servers and making use of the relation between the Erlang B and the Erlang C functions. The approach leads to a straightforward algebraic analysis of the properties of the performance measures of interest and avoids cumbersome manipulations