QoS Guarantee in the Erlang Multirate Loss Model based on Derivatives of Blocking Probabilities I. D. Moscholios*, J. S. Vardakas**, M. D. Logothetis** and A. C. Boucouvalas* *Dept. of Telecommunications Science and Technology, University of Peloponnese, Tripoli 221 00, Greece **WCL, Dept. of Electrical & Computer Engineering, University of Patras, 265 04 Patras, Greece. E-mail: idm@uop.gr , jvardakas@wcl.ee.upatras.gr , m-logo@wcl.ee.upatras.gr , acb@uop.gr Abstract - We consider a single-link loss system of capacity C bandwidth units, 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 the bandwidth reservation (BR) policy. The BR policy is used in teletraffic engineering in order to achieve call blocking probability (CBP) equalization among different service classes. Such a single-link loss system has been analytically described by the Erlang Multirate Loss Model 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 with respect to offered traffic-load of any service-class under the BR policy. We further show through an analytical formula how CBP derivatives can be used to determine approximate CBP when small variations of offered traffic-load are considered. Keywords: loss system; call blocking; derivatives; erlang; reservation; I. INTRODUCTION The existence of service-classes with different bandwidth per call requirements motivates several bandwidth sharing policies, at call-level, in order for a link to offer a certain quality of service (QoS) to each service-class. The most common policies are the complete sharing (CS) [1], [3] and the bandwidth reservation (BR) policy [2]. In the CS policy, calls with higher bandwidth requirements receive worse QoS, i.e. call blocking probability (CBP), than calls with less bandwidth requirements. In the BR policy, a fraction of the available link bandwidth is reserved to benefit calls with higher bandwidth requirements and therefore CBP equalization among calls of different service-classes can be achieved via a proper selection of such a fraction. The basis in the analysis of call blocking behaviour of a link accommodating K service-classes under the BR policy is the Erlang Multirate Loss Model (EMLM/BR) [4]. A call of service-class k (k=1,…,K) arrives to the system according to a Poisson process and is accepted only if the sum of its required bandwidth plus its BR parameter, denoted as t(k), is less or equal to the available link bandwidth. In that case the call remains in the system for an exponentially distributed service time. Otherwise the call is blocked and lost (no waiting room is assumed). Note that the BR parameter of a service-class k, t(k), expresses the bandwidth units (b.u.) reserved to benefit calls of service-classes other than k. In order to calculate CBP in the EMLM/BR, Roberts proposed in [4] an approximate but recursive formula for the calculation of the link occupancy distribution. The approximate nature of Roberts’ formula is based on the fact that the steady state distribution of the number of calls in the link does not have a product form solution (PFS) [2]. If the BR parameter of each service-classes k, t(k), is zero, then the EMLM results and the corresponding bandwidth allocation policy is the CS policy. In the EMLM, exploiting the fact that the steady state distribution of the number of calls in the link has a PFS [2], an accurate recursive formula (known as Kaufman-Roberts formula, KR formula) has been separately proposed by Kaufman [1] and Roberts [3] which determines the link occupancy distribution and simplifies the CBP determination. This simplification resulted in a large amount of extensions of the EMLM and applications of the KR formula both in wired (e.g.[5]-[8]) and wireless networks (e.g. [9]-[12]). In this paper we consider the EMLM/BR and adopt Roberts’ formula [4] as a springboard to the calculation of CBP derivatives with respect to offered traffic-load. The determination of CBP derivatives with respect to offered traffic-load is significant in multirate loss systems since it enables the study of the interaction between different service classes that share the same system. Having determined the CBP derivatives with respect to offered traffic-load, we can approximate CBP for small variations of offered traffic-load by a formula already proposed for the EMLM in [13]. The remainder of this paper is as follows: In section II we review the EMLM/BR. In section III we propose an algorithm for the calculation of CBP derivatives (with respect to offered traffic-load) in the EMLM/BR. In section IV we present a formula that TMTE-2 978-1-86135-369-6/10/$25.00 ©2010 IEEE 822 CSNDSP 2010