# Probability and Statistics Seminar

### A Stochastic Model for Throughput in Wireless Data Networks with Single-Frequency Operation and Partial Connectivity under Contention-Based Multiple Access Modes

Wireless data (packet) networks operating in a common frequency channel, as happens for example with a Basic Service Set in the IEEE 802.11 (WiFi) system, are subject to intrinsic impairments. One such major impairment results from the broadcast nature of the radio channel: if two different transmissions arrive at a receiver with any time overlap, they will interfere destructively and thus one or both of the corresponding packets will not be correctly received (packet collision), thus wasting radio channel transmission time and possibly requiring a retransmission of the original packet(s).

In order to achieve a better utilization of the scarce radio channel resource, stations in wireless networks use multiple acess algorithms to attempt to usefully coordinate their radio transmissions. One example is given by Carrier Sense Multiple Access (CSMA), used as a basis for the sharing of the radio channel in the WiFi system, which establishes that a station should not start a new packet transmission if it can hear any other station transmitting. In a network with radio connectivity between any pair of stations (fully connected) and negligible propagation delays, such algorithm succeeds at completely preventing the existence of collisions. That is however not the case if there exist pairs of stations that cannot directly hear each other (partial connectivity). Many other multiple access algorithms have been proposed and studied.

In this talk will be presented a stochastic model for the study of throughput (i.e., the long-term fraction of time that the radio channel is occupied with successful packet transmissions) in the class of networks described above. The talk will start with a short description of the communication functions and structure of the system under study and of the class of multiple access algorithms considered. Following will be presented a Markovian model for the representation of the time evolution of the packet transmissions taking place in the network and a result given on the existence of a product form for its stationary probabilities. The next step will be to show how the desired throughputs can be obtained from the steady-state probabilities of this process and the average durations of the successful packet transmissions. The latter are obtained from the times to absorption in a set of auxiliary (absorbing) derived Markov chains. Finally, and time permitting, a reference will be made to results concerning the insensitivity of product form steady state solutions, when they exist, to the distribution of packet lengths and retransmission time intervals, by means of a Generalized Semi-Markov Process representation.