  The Erlang loss formula, also known as the Erlang B formula, has been known
for over a century and has been used in a wide range of applications, from
telephony to hospital intensive care unit management. It provides the blocking
probability of arriving customers to a loss system involving a finite number of
servers without a waiting room. Because of the need to introduce priorities in
many services, an extension of the Erlang B formula to the case of a loss
system with preemptive priority is valuable and essential. This paper
analytically establishes the consistency between the global balance (steady
state) equations for a loss system with preemptive priorities and a known
result obtained using traffic loss arguments for the same problem. This paper,
for the first time, derives this known result directly from the global balance
equations based on the relevant multidimensional Markov chain. The paper also
addresses the question of whether or not the well-known insensitivity property
of the Erlang loss system is also applicable to the case of a loss system with
preemptive priorities, provides explanations, and demonstrates through
simulations that, except for the blocking probability of the highest priority
customers, the blocking probabilities of the other customers are sensitive to
the holding time distributions and that a higher variance of the service time
distribution leads to a lower blocking probability of the lower priority
traffic.
