Analysis of phase transition phenomenon in packet networks

Citation
M. Mandjes et Jh. Kim, Analysis of phase transition phenomenon in packet networks, ADV APPL P, 33(1), 2001, pp. 260-280
Citations number
27
Categorie Soggetti
Mathematics
Journal title
ADVANCES IN APPLIED PROBABILITY
ISSN journal
00018678 → ACNP
Volume
33
Issue
1
Year of publication
2001
Pages
260 - 280
Database
ISI
SICI code
0001-8678(200103)33:1<260:AOPTPI>2.0.ZU;2-G
Abstract
The multiplexing of variable bit rate traffic streams in a packet network g ives rise to two types of queueing. On a small time-scale, the rates at whi ch the sources send is more or less constant, but there is queueing due to simultaneous packet arrivals (packet-level effect). On a somewhat larger ti me-scale, queueing is the result of a relatively high number of sources sen ding at a rate that is higher than their average rate (burst-level effect). This paper explores these effects. In particular, we give asymptotics of t he overflow probability in the combined packet/burst scale model. It is sho wn that there is a specific size of the buffer(i.e. the 'critical buffer si ze') below which packet-scale effects are dominant, and above which burst-s cale effects essentially determine the performance-strikingly, there is a s harp demarcation: the so-called 'phase transition'. The results are asympto tic in the number of sources n. We scale buffer space B and link rate C by n, to nb and nc, respectively; then we let n grow large. Applying large dev iations theory we show that in this regime the overflow probability decays exponentially in the number of sources n. For small buffers the correspondi ng decay rate can be calculated explicitly, for large buffers we derive an asymptote (linear in b). The results for small and large buffers give rise to an approximation for the decay rate (with general b), as well as for the critical buffer size. A numerical example (multiplexing of voice streams) confirms the accuracy of these approximations.