Derive the value of p that maximizes the efficiency equation. 2. Please show the derivation. P8. 1 0 <> Discussions. Post Discussion. If we find the derivative of this expression,… Tweet. b. (6 pts.) exercises e-3 Aloha & Slotted Aloha efficiency Suppose N stations have packets to send meach transmits in slot with probability p mprob. Graph the efficiency of slotted ALOHA and pure ALOHA as a function of p for the following values of N: a. N#15. Post the discussion to improve the above solution. Slotted Aloha efficiency limit of Nq*(1-q*)N nodes with many frames to send, each q (new arrival or re-Tx) used for useful probthat node 1 has success in a slot = q(1-q)N-1 Total expected utilization = Nq(1-q)N-1 For max efficiency with N nodes, find q* that maximizes Nq(1-q)N-1 For many nodes, take N-1 as N goes to infinity, gives 1/e = .37 Find the value of p that maximizes this expression. Slotted Aloha is an improvement to the original Aloha protocol, where discrete time slots were introduced to increase the maximum throughput while reducing collisions. Answer. Recall that when there are N active nodes, the efficiency of slotted ALOHA is Np(1 – p) N–1 . Hint: (1 – 1/N) N approaches 1/e as N approaches infinity. In Pure ALOHA no slotting was done but the efficiency was poor. This can reduce the efficiency of a network since colliding frames will cause loss of data in both the frames. My question is, if only one station is transmitting in one time slot, then there will be no collisions and since we are talking about maximum efficiency, all time slots will be utilized. For a large number of users the ALOHA protocol is considered. In Slotted ALOHA, slots have been made, so that every frame transmission starts at the beginning of the slot and throughput is increased by a factor of 2. The only condition is that station must start its transmission from the beginning of the time slot. Slotted Aloha- Slotted Aloha divides the time of shared channel into discrete intervals called as time slots. Engineering. Slotted-Aloha type MAC protocols don't perform carrier sensing and synchronize the transmissions into time-slots. There are two versions of ALOHA that is Pure ALOHA and Slotted ALOHA. a. a) Recall that when there are N active nodes, the efficiency of slotted ALOHA is Np(1-p)^(N-1). successful transmission S is: by single node: S= p (1-p)(N-1) by any of N nodes S = Prob (only one transmits)= N p (1-p)(N-1) Suppose there are N active nodes sharing the channel. Aloha does not avoid hidden or exposed nodes and is generally inefficient in terms of usage of the channel: pure Aloha leads to a maximum efficiency of less than 1/(2e) ≈ 18.5%, compared to a single user system. https://www.cs.umd.edu/~shankar/417-F01/Slides/chapter5a-aus/sld022.htm c. N#35. Slotted Aloha can utilize the channel up to nearly 1/e ≈ 37%. b. N#25. The maximum efficiency of Pure Aloha is very less due to large number of collisions. Find the value of p that maximizes this expression. Any station can transmit its data in any time slot. Recall that the efficiency is therefore Np(1-p)N-1, a. Using the value of p found in (a), find the efficiency of slotted ALOHA by letting N approach infinity. Consider a slotted ALOHA system. b. In slotted ALOHA, the maximum efficiency is 36.8% when only one station transmits in one time slot . Eytan Modiano Slide 10 Throughput of Slotted Aloha • The throughput is the fraction of slots that contain a successful transmission = P(success) = g(n)e-g(n) – When system is stable throughput must also equal the external arrival rate (λ) – What value of g(n) maximizes throughput? The efficiency of slotted ALOHA divides the time slot from the beginning of the time of channel! 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