Analysis of Aloha and Slotted Aloha
Jie Gao
∗
September 25, 2009
1 Aloha and Slotted Aloha
Aloha is a time-division multiple access scheme. A user, whenever has data to transmit, immediately
transmits, irrespective of whether the other users use the channel or not. In Slotted Aloha, time is
divided into time slots. A user with data will transmit at the beginning of the next time slot.
In this note we analyze the throughput of the system. The throughput is defined as the ratio
of the number of packets delivered successfully and the total number of packets possible (assuming
perfect global coordination). The throughput measures the efficiency of the system.
1.1 Assumptions
In the following analysis we make the following assumptions.
• There are total number of N users in the system.
• All packets have length T .
• Each user transmits with probability p within a time period of T .
Now the average number of packets in the system within a period of time T is simply λ = N · p.
The average number of packets in the system within a period of time 2T is λ
0
= 2N · p. The arrival
of independent events with a rate of λ within unit time is modeled by the Poisson distribution:
P (k ) =
λ
k
e
−λ
k!
,
in which λ is the rate, the average numb er of packet transmissions within unit time T , and P (k) is
the probability that k packet transmissions occur within unit time T .
1.2 Vulnerable periods
We use the term vulnerable period to denote the period of time that a packet can possibly collide
with a given packet x. In Aloha system, the vulnerable perio d has length 2T —- any packet that
starts within time T from the starting point of x (either before or after) will collide with x. In
slotted Aloha, two packets either collide completely or do not overlap at all. The vulnerable period
is reduced to T — the slot before the transmission of packet x.
∗
Department of Computer Science, Stony Brook University, Stony Brook, NY 11794, USA, [email protected].
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