1.1 Polling network analysis

A polling network is a computer communications network that uses polling to control access to the network. Each node or station on the network is given exclusive access to the networkin a predetermined order. Permission to transmit on the network is passed from station to station using a special message called a poll . Polling may be centralized (often called hub polling ) or decentralized ( distributed ). In hub polling, the polling order is maintained by a single central station or hub . When a station finishes its turn transmitting, it sends a message tothe hub, which then forwards the poll to the next station in the polling sequence. In a decentralized polling scheme, each stationknows its successor in the polling sequence and send the poll directly to that station. To simplify matters, we will assume adistributed polling scheme.

The analysis of a polling network uses the results of the analysis of an M/G/1 queue with vacations. Each vacationcorresponds to the transfer of the poll from one station to the next in the polling cycle. We divide time into alternatingtypes of intervals: polling intervals , during which the poll is transferred between stations, and transmission intervals , during which the station with the poll transmits packets.

Polling networks come in three flavors: gated, exhaustive, and partially gated. In a gated system, each station is allowed to transmit only those packets that arrived prior tothe start of the poll interval (i.e., prior to the start of the vacation preceding the station's use of the network). An exhaustive scheme allows a station to transmit any packets that arrive before it transfers the poll to the nextstation. A partially gated network allows stations to transfer all packets that arrive by the time the poll does.Polling networks will typically be partially gated or exhaustive, not gated.

We assume that arrivals at each of the $m$ stations are independent Poisson processes with rate $\frac{\lambda }{m}$ . Note that "arrival" refers to a message arriving from the "outside world" to a station in order to be transmittedover the network; it does not mean the arrival of a message that has been transmitted over the network.

Exhaustive system

${\alpha }_{jk}$ is the expected value of $Y_i$ given that packet $i$ arrives in user $j$ 's polling or data interval and belongs to user $(j+k)\mod m$ . $${\alpha }_{jk}=\begin{cases}0 & \text{if $k=0$}\\ \langle V_{(j+1)modm}\rangle +\dots +\langle V_{(j+k)modm}\rangle & \text{if $k> 0$}\end{cases}$$ We first remove the condition that the packet belongs to station $(j+k)\mod m$ by assuming that a packet belongs to a particular station with probability $\frac{1}{m}$ for all stations. The expected value of $Y_i$ given that packet $i$ arrives in user $j$ 's polling or data interval is given by $$\frac{1}{m}\sum_{k=1}^{m-1} {\alpha }_{jk}=\sum_{k=1}^{m-1} \frac{m-k}{m}\langle V_{(j+k)modm}\rangle$$ Since all users are identical, they have equal average length data intervals in steady-state, and the steady-stateprobability that a packet arrives in a particular user's data interval is $\frac{\rho }{m}$ . Similarly, the probability that a packet arrives during a particular user's polling interval is $\frac{(1-\rho )\langle V_r\rangle }{\sum_{k=0}^{m-1} \langle V_k\rangle }$ .

Partially gated system

In the partially gated system, a packet that arrives during a user's own data interval is delayed by an additional $m\langle V\rangle$ on average, and this occurs with probability $\frac{\rho }{m}$ , thus increasing $Y$ by $\rho \langle V\rangle$ compared to the exhaustive case. $$W=\frac{\lambda \langle X^2\rangle }{2(1-\rho )}+\frac{(m+\rho )\langle V\rangle }{2(1-\rho )}+\frac{{\sigma }_V^2}{2\langle V\rangle }$$

Fully gated system

If the system is fully gated, then a packet that arrives during a user's own polling interval is also delayed by anaverage of $m\langle V\rangle$ , and this is in addition to the extra delay incurred in the partially gated case. The probability of thisoccurring is $\frac{1-\rho }{m}$ , thus increasing $Y$ by $(1-\rho )\langle V\rangle$ compared to the partially gated case. $$W=\frac{\lambda \langle X^2\rangle }{2(1-\rho )}+\frac{(m+2-\rho )\langle V\rangle }{2(1-\rho )}+\frac{{\sigma }_V^2}{2\langle V\rangle }$$