# Probabilistic Retry and Threshold Multirate Loss Models for Impatient Calls
## 1 Introduction  
Multirate loss models (MLMs) are widely adopted in the literature for the determination of call blocking probabilities (CBP), a significant performance index in contemporary multiservice networks [1, 2]. Among the MLMs, the basis is considered to be the Erlang MLM (EMLM) not only because CBP can be efficiently determined via the Kaufman-Roberts recursive formula [3, 4] but also due to the fact that it has been widely adopted for the CBP computation in various types of networks (from wired [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], to optical [17, 18, 19, 20], wireless [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] and even satellite networks [35, 36, 37, 38]). The EMLM is used for the analysis of a link of certain capacity (in terms of bandwidth units (b.u.)) that services calls of different service-classes with various bandwidth requirements. The call arrival process in the link is Poisson while the available b.u. are shared via the complete sharing policy. According to this policy, a new call will be accepted in the link if there exist available b.u. at the time of the call's arrival. Otherwise call blocking occurs. An accepted call remains in the link for a generally distributed service time. The fact that the steady state probabilities in the EMLM have a product form solution (PFS) is significant for the derivation of the classical Kaufman-Roberts formula, which leads to the accurate (compared to simulation) CBP computation [3, 4]. In [39, 40], two extensions of the EMLM are proposed. In the first extension, blocked calls may immediately retry to be accepted in the link with a reduced b.u. requirement and an increased service time. In that case, we have a retry loss model. Call blocking occurs if the reduced b.u. are not available. In the second extension, the b.u. requirement of a new call depends on the occupied link bandwidth. If the occupied link bandwidth is above a threshold (which is common for all service-classes), then the new call tries immediately to be accepted in the link with a reduced b.u. requirement and consequently an increased service time. In that case, we have a threshold loss model. Call blocking occurs if the reduced b.u. are not available. Conceptually, the main difference between the two loss models is that in a threshold loss model, a new call does not have to be blocked with itsinitial bandwidth requirement in order to retry with a reduced value. It is the value of the threshold and the occupied link bandwidth that decide with which bandwidth requirement (the initial or the reduced), a new call will try to be accepted in the link. Note that in both models, calls may have elastic bandwidth requirements but fixed bandwidth allocation (the initial or the reduced bandwidth requirement) during their service time. Because of this, we classify the models of [39, 40] as models that consider stream traffic. This is different from the case of elastic traffic where the bandwidth of in-service calls fluctuates during their service time [2, 13]. In this paper, we focus on both loss models and study the case where a call may wish to enter a system with a reduced b.u. requirement and consequently increased service time, with a predetermined probability. Such a probability expresses the fact that the increased service time may cause impatience to calls and therefore may decide not to enter a system, although the reduced b.u. are available. The steady-state probabilities in both loss models cannot be described via a PFS due to the existence of retry (in the retry loss model), the existence of the threshold (in the threshold loss model), and the incorporation of impatience (in both models). However, we propose approximate formulas for the recursive computation of the link occupancy distribution and consequently CBP whose accuracy is validated via simulation.