probability, that two or more sensors have data to send in the same frame. If exactly two out of the sensors, Zoya and Yoshi, have data to send in the same frame, then the following outcomes are possible:
1 Case 1. Each sensor picks a different reservation slot. Then Basil receives both reservation packets and decides to allocate the data slot to, for example, Zoya. Yoshi tries again to send its reservation packet in a future frame.
2 Case 2. Both sensors pick the same reservation slot and end up in a collision. Then Basil cannot allocate the data slot to any of the two sensors, leaving the data slot empty.
Each of the two outcomes occurs with probability and in both cases it becomes apparent that having a fixed is not efficient. In the first case, the successful reservation of Yoshi is wasted1. In the second case, the data transmission slot is empty and wasted, as none of the two sensors can use it for transmission.
This leads us to think of a more efficient solution: does not need to be fixed, but it would be the best if the value of can be adapted to be equal to the number of successful outcomes, denoted by , where in the reservation frame of size . Basil needs to dynamically set , since in each new frame is a random number. Recalling the discussion from the previous chapter, this flexibility demands additional signaling information, as Basil needs to decide the value of after the reservation phase is finished and then communicate the value to the terminals. Since there can be at most successful reservations, the number of data slots for a frame can range from 0 to and this number can be specified in the allocation packet, along with the addresses of the devices to which the slots are allocated.
The essence of the described scheme is to allow all the users to randomly access the reservation slots. This method of random access is known in the literature as framed ALOHA, as it is a variant of the basic ALOHA protocol. The next question is: how do we choose the number of reservation slots ? We will carry out a quick, non-rigorous analysis, in order to get an insight into the design choices for the described type of system.
2.1.1 Randomization that Maximizes the ALOHA Throughput
The question of choosing the optimal cannot be answered without providing additional elements of the model in which random access is used. To start with, we have not specified the random process that describes the way the sensors attempt to send their packets. The way to model this is to assume a random process that describes whether a sensor device has something to transmit in a given frame. In order to shed light on these issues, we can formulate a simpler problem that is still relevant for making the optimized choice of . Let there be active sensors with data to send among the population of sensors. Each active sensor is trying to send a reservation packet in one of the slots. It is important to state that the sensors are not mutually coordinated in any way before starting the random access process.
Having said that, there is a certain (dark room) symmetry in the problem: all the sensors look equal to the receiver and each of the reservation slots looks equal to each sensor. This means that, if a particular sensor Zoya needs to pick a single reservation slot, then each of the reservation slots should have an equal chance to be picked, with probability . Considering this, the probability that Zoya will have a successful transmission of her reservation packet in a particular slot is
(2.1)
which is the probability that Zoya sends in that slot and that none of the other sensors chose it for transmission.
The probability that there is a successful transmission in that slot by any of the sensors is:
(2.2)
It can be shown that the latter expression is maximized when . Hence, the best way is to choose the number of reservation slots to be equal to the number of active sensors that are contending via random access (framed ALOHA), such that the probability of successful reception in a given slot is:
(2.3)
Clearly, this requires knowledge of the number of active sensors in the total population of sensors.
Let us see the implications that this has on our system. Before the frame starts, Basil knows that there will be sensors that will be contending with each other in order to request access, but he does not know the identities of these sensors. Note that this is the crucial assumption in the problem setup for random access, since if Basil knows which sensors will require access, then there is no need for randomized contention: namely, Basil can simply set and allocate one data slot to each sensor. Although this observation seems trivial, it is very
probability, that two or more sensors have data to send in the same frame. If exactly two out of the 