{'fraction_non_alphanumeric': 0.030710172744721688, 'fraction_numerical': 0.0, 'mean_word_length': 5.181818181818182, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
New algorithms for Performance Measures Derivatives in the Erlang Multirate Loss Model including the Bandwidth Reservation Policy

We consider a single-link loss system with fixed bandwidth capacity accommodating K service-classes of Poisson traffic with different bandwidth-per-call requirements. Calls of all service-classes compete for the available link bandwidth, under two different policies: a) the complete sharing (CS) policy and b) the bandwidth/trunk reservation (BR) policy. In the CS policy, a call is blocked and lost if and only if the available link bandwidth is less than the call’s bandwidth requirement. Such a single-link loss system has been analytically described by the Erlang Multirate Loss Model (EMLM). On the other hand, the BR policy is used in order to achieve call blocking probability (CBP) equalization among service classes, or, alternatively, guarantee a certain quality of service for each service-class. Such a single-link loss system has been analytically described by the EMLM under the BR policy (EMLM/BR). In this paper, we focus on the problem of determining, in an efficient analytical way, derivatives of blocking probabilities, throughput rates and average number of calls in the system, with respect to offered traffic-load, arrival rate and service rate of any service-class under both the EMLM and EMLM/BR models. We further show through analytical formulas how derivatives can be used to successfully approximate the above mentioned performance measures when small variations of offered traffic-load are considered.