rel="nofollow" href="#fb3_img_img_431a6d94-d421-53de-963b-7db20743bd79.png" alt="images"/> 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 infinite, and each new request comes from a new sensor, which also means that each sensor has only one request. Such a hypothetical scenario represents the most difficult case for coordination among the sensors. In the following we provide the reasoning behind the choice of the infinite-size sensor population.
Instead of active sensors, each with a single request, we consider sensors, where each sensor has packets. The total number of packets to be sent in the system is , which makes the overall traffic load equal to the case with single-packet users. The following protocol is run by each sensor. The sensor Zoya applies the framed ALOHA protocol until it successfully sends her first request. After succeeding, Zoya records (a) the number of sensors that sent their first requests successfully before Zoya, which she learns from Basil's feedback; (b) puts on hold her access until the remaining sensors have sent their first requests successfully. Note that, after this randomized contention is finalized, Zoya has a unique number , where . Since every sensor applies the same protocol, each sensor has a unique token, which is a number between 1 and . After contending to send the first request and obtaining the token, the sensors no longer need to contend, but they are served through a TDMA frame with slots, where, for example, the slot number is allocated to Zoya. This is reminiscent of the use of random access as a technique for initial access, after which the transmissions are coordinated and scheduled.
When there are sensors with a single request each and goes to infinity, the system throughput is packets per slot, since the sensors need to contend indefinitely. Let
