“A non parametric estimate of performance in queueing models with long-range correlation, with applications to telecommunication” Pier Luigi Conti, Università La Sapienza (Roma) Livia De Giovanni, Università del Molise

The Problem Broadband telecommunication networks are based on packet switching technique The information stream produced by a terminal equipment is packetized and delivered to destination in the form of fixed or variable length packets on the basis of a label added to each packet identifying the virtual connection source/destination The ATM (Asynchronous Transfer Mode) technique is based on fixed length cells (53 byte). IP protocol is based on variable length datagrams. Opposite to ATM networks, IP networks operate without preventive traffic control funcions ATM cells are routed on the basis of the label from source to destination through ATM nodes, which interconnect input links with output links. Output links are equipped with buffers to store and schedule cells for transmission

The Problem Cell Loss Probability (Cell Loss Ratio CLR) is one of the most important performance measure/Quality of Service (QoS) parameter in ATM and/or IP networks In IP networks using ATM as a transport technique, for instance, the loss of ATM cells and consequently of IP datagrams leads to retransmission which may cause congestion and delay The Cell Loss Probability with respect to a finite buffer is defined as the “long term fraction of lost cells” (because they find the buffer full). As an approximation of CLR the overflow probability is used, defined with respect to an infinte buffer as the probability that the number of cells in the buffer exceeds a cartain threshold

The problem The Long Range Dependence (LRD) nature of the arrival process has a negative impact on the overflow probability (Addie (1999)) It is then of crucial importance: to validate the presence of LRD in the arrival process (of ATM cells) to define an estimate of the overflow probability robust with respect to the LRD nature of the arrival process

The model G/G/1 with infinite queue The time is divided into consecutive intervals (time slots) corresponding to the ATM cell transmission time at the ATM link bandwidth Let V n be the random variable (r.v.) “unfinished work” of the queue at the beginning of the n-th interval A n : r.v. repesenting the number of cell arrivals in a time slot B n : r.v. repesenting the work that can be processed in single time slot(B n = 1) Y n = A n - B n

The overflow probability Under the following assumptions: i) {Y n, n  1} is a stationary Gaussian process ii) Var(Y 1 +Y 2 +…+Y n ) = n 2H  2 1/2  H <1 (the arrival process is LRD with Hurst parameter H) it holds (Addie 1999): m: E(Y n ); H: Hurst parameter;    Var(Y n ); I (.) : indicator function

Asymptotic results The results hold true without assuming that the model is stable (m<0; m=  -1,  = E(A n )/E(B n )) Tha basic idea is to consider estimates of m, H e  2 and then to replace them in Addie’s formula to get a plug-in estimate of the overflow probability As estimator of the vector (H,  2 ) we take here the aggregated Whittle’s approximate maximum likelihood estimator. For stationary Gaussian processes the estimator is n 1/2 -consistent, efficient and asymptotically Gaussian (Fox (1986), Dahlhaus (1989)) provided that in the likelihood function m is replaced by a n 1-H -consistent estimate of m (for instance the sample mean) To obtain a semiparametric estimate the vector is estimated for the aggregated process

The estimate of the overflow probability The estimate of the overflow probability that follows is:

Asymptotic results Proposition 1: Proposition 2: Proposition 3:

Confidence Intervals Proposition 3 suggests to adopt the following confidence interval for A: Proposizion 4: the asymptotic size of the confidence interval is 1-  if m 0.

The data The ATM network under consideration is the experimental European ATM network set up by a large group of European network operators (JAMES Joint ATM ExperimentS project) The applications considered are videoconference (application A), transport of routing information between network routers (application B) and teleteaching (application C). They use the Internet Protocol over ATM The input process A n has been obtained, slot by slot, as the number of cells requiring transmission in the same time slot. The service time process B n is constant and equal to 1

Confidence Intervals (0.65  0.13) (0.64  0.13) (0.65  0.13) Confidence Intervals for H, (1-  = 95%

Confidence Intervals  =0.2 (6.*  3. * ) (2.2*  1.2* )  =0.57 (3.*10 -2  9. * ) (9. *10 -5  1.2* )  =0.78 (3.*10 -1  1.5* ) (8. *10 -2  2. * ) Confidence Intervals for A, (1-  = 95%

Conclusion The previous analysis shows that: the Hurst parameter is significantly different from zero the presence of LRD does have a negative impact on the (estimated) overflow probability the Cell Loss Probability QoS requirement (3*10 -7 ; ) estimated by the overflow probability is not satisfied for relevant values of  (0.57, 0.78) independently of the buffer size

INOUT conversione LABEL Routing Tag BUFFER