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Unformatted text preview: in slotted Aloha with
p = 1/N . When N is large, these numbers are 1/e and 1 − 2/e ≈ 26%, respectively.
It is interesting that the number of idle slots is the same as the utilization: if we
increase p to reduce the number of idle slots, we don’t increase the utilization but
actually increase the collision rate.
4. Stabilization is crucial to making Aloha practical. We studied a scheme that adjusts
the transmission probability, reducing it multiplicatively when a collision occurs and
increasing it (either multiplicatively or to a ﬁxed maximum value) when a successful
transmission occurs. The idea is to try to converge to the optimum value.
5. A non-zero lower bound on the transmission probability is important if we want
to improve fairness, in particular to prevent some nodes from being starved. An
upper bound smaller than 1 improves fairness over shorter time scales by alleviating
the capture effect, a situation where one or a small number of nodes capture all the
transmission attempts for many time slots in succession.
6. Slotted Aloha has double the utilization of unslotted Aloha when the number of
backlogged nodes grows. The intuitive reason is that if two packets are destined to
collide, the “window of vulnerability” is larger in the unslotted case by a factor of
7. A broadcast network that uses packets that are multiple slots in length (i.e., mimicking the unslotted case) can use carrier sense if the medium is a true broadcast
medium (or approximately so). In a true broadcast medium, all nodes can hear each
other reliably, so they can sense the carrier before transmitting their own packets. By
“listening before transmitting” and setting the transmission probability using stabilization, they can reduce the number of collisions and increase utilization, but it is
hard (if not impossible) to eliminate all collisions. Fairness still requires bounds on
the transmission probability as before.
8. With a contention window, one can make the transmissions from backlogged nodes
occur according to a uniform distribution, instead of the geometric distribution imposed by the “send with probability p” schemes. A uniform distribution in a ﬁnite
window guarantees that each node will attempt a transmission within some ﬁxed
number of slots, which is not true of the geometric distribution. Acknowledgments Mythili Vutukuru provided several useful comments that improved the explanations presented here. Thanks also to Sari Canelake and Lavanya Sharan for suggesting helpful
improvements. SECTION 10.9. SUMMARY 17 Problems and Questions These questions are to help you improve your understanding of the concepts discussed in
this lecture. The ones marked *PSet* are in the online problem set.
1. In the Aloha stabilization protocols we studied, when a node experiences a collision,
it decreases its transmission probability, but sets a lower bound, pmin . When it transmits successfully, it increases its transmission probability, but sets an upper bound,
(a) Why would we set a lower bound on pmin that is not too close to 0?
(b) Why would we set pmax to be signiﬁcantly smaller than 1?
(c) Let N be the average number of backlogged nodes. What happens if we set
pmin >> 1/N ?
2. *PSet* Alyssa and Ben are all on a shared medium wireless network running a variant of slotted Aloha (all packets are the same size and each packet ﬁts in one slot).
Their computers are conﬁgured such that Alyssa is 1.5 times as likely to send a packet
as Ben. Assume that both computers are backlogged.
(a) For Alyssa and Ben, what is their probability of transmission such that the utilization of their network is maximized?
(b) What is the maximum utilization?
3. *PSet* You have two computers, A and B, sharing a wireless network in your room.
The network runs the slotted Aloha protocol with equal-sized packets. You want B
to get twice the throughput over the wireless network as A whenever both nodes are
backlogged. You conﬁgure A to send packets with probability p. What should you
set the transmission probability of B to, in order to achieve your throughput goal?
4. *PSet* Ben Bitdiddle sets up a shared medium wireless network with one access
point and N client nodes. Assume that the N client nodes are backlogged, each with
packets destined for the access point. The access point is also backlogged, with each
of its packets destined for some client. The network uses slotted Aloha with each
packet ﬁtting exactly in one slot. Recall that each backlogged node in Aloha sends
a packet with some probability p. Two or more distinct nodes (whether client or
access point) sending in the same slot causes a collision. Ben sets the transmission
probability, p, of each client node to 1/N and sets the transmission probability of the
access point to a value pa .
(a) What is the utilization of the network in terms of N and pa ?
(b) Suppose N is large. What val...
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- Fall '13
- The Land