Editing Exponential backoff (section)

===Stability and adaptive backoff===

To understand stability, Lam created a discrete-time [[Markov chain]] model for analyzing the statistical behaviour of slotted ALOHA in chapter 5 of his dissertation.<ref name="Lam74">{{cite thesis|last=Lam |first=Simon S. |date=March 1974 |title= Packet Switching in a Multi-Access Broadcast Channel with Application to Satellite Communication in a Computer Network, Ph.D. dissertation, 306 pages |url= https://www.cs.utexas.edu/users/lam/Vita/UCLA/| institution= UCLA-ENG-7429 (ARPA), UCLA School of Engineering and Applied Science }}</ref> The model has three parameters: {{var|N}}, {{var|s}}, and {{var|p}}.  {{var|N}} is the total number of users. At any time, each user may be idle or blocked.  Each user has at most one packet to transmit in the next time slot. An idle user generates a new packet with probability {{var|s}} and transmits it in the next time slot immediately. A blocked user transmits its backlogged packet with probability {{var|p}}, where 1/{{var|p}} = ({{var|K}}+1)/2 to keep the average retransmission interval the same. The throughput-delay results of the two retransmission methods were compared by extensive simulations and found to be essentially the same.<ref>Fig. 5-1 on page 100, Chapter 5, in Lam’s dissertation</ref>

Lam’s model provides mathematically rigorous answers to the stability questions of slotted ALOHA, as well as an efficient algorithm for computing the throughput-delay performance for any stable system. There are 3 key results, shown below, from Lam’s Markov chain model in Chapter 5 of his dissertation (also published jointly with Professor Len Kleinrock, in IEEE Transactions on Communications.<ref name="KL75">{{cite journal |last1=Kleinrock |first1=Leonard |last2=Lam S.|first2=Simon |date=April 1975 |title=Packet-Switching in a Multi-Access Broadcast Channel: Performance Evaluation|journal=IEEE Transactions on Communications|url=https://www.cs.utexas.edu/users/lam/Vita/Jpapers/KleinrockLam75.pdf |volume=COM-23|number=4|pages=410–423|doi=10.1109/TCOM.1975.1092814 |accessdate=16 February 2023}}</ref>)
#  Slotted ALOHA with Poisson arrivals (i.e., infinite {{var|N}}) is inherently unstable, because a stationary probability distribution does not exist. (Reaching steady state was a key assumption used in the models of Abramson and Roberts.)
#  For slotted ALOHA with a finite {{var|N}} and a finite {{var|K}}, the Markov chain model can be used to determine whether the system is stable or unstable for a given input rate ({{var|N}}×{{var|s}}) and, if it is stable, compute its average packet delay and channel throughput rate. 
# Increasing {{var|K}} increases the maximum number of users that can be accommodated by a stable slotted ALOHA channel.<ref>Fig. 5-9 on page 114 in Chapter 5 of Lam's dissertation, or Fig. 10 on page 418 in the 1975 Kleinrock-Lam paper.</ref>
====Corollary====
For a finite ({{var|N}}{{times}}{{var|s}}), an unstable channel for the current {{var|K}} value can be made stable by increasing {{var|K}} to a sufficiently large value, to be referred to as its {{var|K}}({{var|N}},{{var|s}}).<ref>Fig 5-10 on page 116 in Chapter 5 of Lam’s dissertation, or Figure 11 on page 418 in the 1975 Kleinrock-Lam paper.</ref>