often overlooked when random access is considered in practical systems. Basil sets , but the number of sensors that successfully send reservation requests is random. This implies that the number of allocated slots is also random. The expected value of is , and one can use (1.8) to calculate the expected goodput in a frame.
Another observation is that decreases with and it reaches as the number of users goes to infinity. The engineering insight from is that, when the users are contending in smaller groups, then the probability of successful transmission experienced by an individual user is higher.
The assumption that Basil knows the exact value of is rather artificial. On the other hand, Basil may know some statistics about the random process according to which the sensors send reservation requests. In that case, it can be reasonable to conclude that Basil knows the expected value of . Although not mathematically rigorous, Basil can work with the expected value as if it is the exact value and use the following approach. At the start of th frame the expected number of sensors that require access, denoted by , is given by:
where is the expected number of new requests generated from new sensors in the previous, th frame. is the expected number of sensors that tried to send request in the previous frames, but did not succeed due to collision. If a frame becomes sufficiently long, then we can apply the law of large numbers, by which the expected values can be approximated as the exact values. For this to be true, the random arrival process of the requests from the sensors should satisfy certain conditions, which we will not discuss in detail here. It suffices to say that, for example, Poisson arrivals of requests over a sufficiently long interval would work. Going back to (2.4), we remove the averaging bar and recast the same equation as . Using the previous analysis on the probability of successful transmission of a request, we can express , but since is large, we can write , which leads to:
Hence, if Basil uses long frames and applies the law of large numbers, then he can have a good guess at the number of contending sensors and practically choose the reservation frame size in an optimal way.
The equation (2.5) can give us further very important insights into the random access protocols. Let us, for a moment, put aside the frame structure considered until now, in which Basil first lets the sensors contend using short reservation frames and then allocates data slots to the successful contenders. Instead, consider the following situation. A very large population of sensors is synchronized to Basil. A periodic frame of data slots and duration of is used, without any additional overhead at the frame start, since all sensors and Basil are assumed to be perfectly synchronized and thus have a perfect knowledge about the moment at which a frame starts. Each sensor that got data to send before the start of the th frame, chooses a random number between 1 and and sends its data in the th slot of the th frame. At the end of the frame, all sensors that sent data successfully receive feedback from Basil. This feedback is assumed to be sent extremely quickly, taking practically zero time. The sensors that did not send the data successfully, treat their data packet as a newly arrived one during the th frame and try again in the th frame. Looking again at the equation (2.5), we can interpret it as follows: if the number of newly arrived requests during each frame of duration is , then this number is equal to the number of successfully sent requests in a frame. Hence, the system is in equilibrium in the sense that each arrived request eventually gets served. Therefore the throughput of this system is, calculated in number of requests (packets) per unit time, is:
(2.6)
Note that, due to the absence of overhead, here the throughput is equal to the goodput. If we take , then the throughput is conveniently expressed in packets per slot and we arrive at the well known formula for maximal throughput of a slotted ALOHA system equal to packets per slot.
However, what does this theoretical value of the ALOHA throughput mean for a practical system? The randomized protocol coordinates the sensor transmissions, such that each sensor eventually transmits its request successfully. The presented analysis captures the following extreme case: the total population of sensors is very large, practically
often overlooked when random access is considered in practical systems. Basil sets 