Splitting Algorithms We have seen that slotted Aloha has maximal - PowerPoint PPT Presentation

Splitting Algorithms We have seen that slotted Aloha has maximal throughput 1 /e Now we will look at more sophisticated collision resolution techniques which have higher achievable throughput These techniques also maintain stability without a complex estimation procedure like in pseudo-Bayesian slotted Aloha The way they obtain this is by choosing different retransmission probability for different nodes, at each time slot during collision resolution the nodes are subdivided into two sets Information Networks – p.1/24

Splitting Algorithms A preliminary explanation as to how this is possible is to consider an algorithm that will make new arrivals wait until an ongoing collision has been resolved Assuming a small attempt rate it is most likely to have only two packets colliding All other nodes will refrain from transmitting until they have observed that those two backlogged packets have been successfully transmitted Each of the colliding packets could then be retransmitted with probability 1 / 2 leading to successful retransmission of one of them with probability 1 / 2 and the other could then be transmitted in next slot Information Networks – p.2/24

Splitting Algorithms With probability 1 / 2 another collision or an idle slot occurs If so, the two packets would again be retransmitted with probability 1 / 2 until a successful transmission occurred which would be followed by the transmission of the remaining packet The probability of two slots for retransmitting the packets is 1 / 2 since this happens if there is no further collision Probability for three slots is 1 / 4 , probability for i slots is 2 − ( i − 1) Information Networks – p.3/24

Splitting Algorithms The expected value for number of slots to send these two packets is thus ∞ 1 i 2 − ( i − 1) = � (1 − 1 / 2) 2 − 1 = 3 i =2 Thus we get a throughput of 2 / 3 for the period during which the collision is resolved Various ways to choose whether to transmit or not in successive slots: Random choice (“flip an unbiased coin”) Using arrival time of its collided packet Node identifier (in case of finite number of nodes) Information Networks – p.4/24

Splitting Algorithms All alternatives have the property that the set of colliding nodes is split into subsets, one of which is transmitting in the next slot If the collision is not resolved, then a further splitting into subsets is performed We assume slotted channel, Poisson arrivals, collision or perfect reception, (0 , 1 , e ) immediate feedback, retransmission of collisions, and an infinite set of nodes When collision occurs in slot k , all nodes not involved in collision goes into waiting mode, the rest are split into two subsets Information Networks – p.5/24

Tree Algorithms First subset transmits in slot k + 1 and if that slot is idle or successful, the second subset transmits in slot k + 2 Alternatively, if another collision occurs in slot k + 1 the first subset is split again and the second subset waits for the resolution of that collision We can describe the splitting procedure with a binary tree, we call the root set of nodes involved in the initial collision S , and the sets it is split up into L (left) and R (right), further splitting of L gives the subsets LL and LR , etc All nodes in the left set and all its descendants (further splitting) is transmitted before the subsets of the right right branch Information Networks – p.6/24

Tree Algorithms From the feedback (0 , 1 , e ) each node can construct the tree and keep track of its own subset in that tree and determine when to retransmit An alternative description is to consider a stack, when collision occurs the set of nodes involved is split into two subsets and these are pushed on the stack, then the subset at head of stack is removed and transmitted (with splitting and pushing on stack if collision occurs) A node with a backlogged packet can keep track of when to retransmit by a counter determining the position of its subset on the stack Information Networks – p.7/24

Tree Algorithms When packet is involved in collision the counter is set to 0 or 1 depending on which subset it is placed in When counter is 0 packet is transmitted If counter is nonzero, it is incremented by 1 for each collision and decremented by 1 for each success or idle A collision resolution period (CRP) is defined to be completed when a success or idle occurs and there are no remaining elements on the stack When a CRP is completed a new CRP is started with the packets that arrived during previous CRP Information Networks – p.8/24

Tree Algorithms If a lot of slots are required during a CRP it is likely that a lot of packets arrived during that time, these will collide and continue to collide until the subsets get small enough in the new CRP The solution is that at the end of a CRP the new arrivals will be split into j subsets, where j is chosen so that the expected number of nodes per subset is slightly greater than 1 These j subsets are placed on the stack and then a new CRP begins Each node will keep track of the number of elements on the stack and the number of slots since the end of previous CRP , the nodes involved in the CRP will also keep track of its position on the stack Information Networks – p.9/24

Tree Algorithms On the completion of a CRP each node determines the expected number of new arrivals and from that a new number j of subsets, and those with new arrivals waiting will randomly choose one of those j subsets and set their counter to the corresponding stack position The maximal throughput available with this algorithm, optimized over the choice of j as a function of expected number of new arrivals, is 0 . 43 packets per slot If a collision is followed by an idle slot, this means that all packets involved in the collision were assigned to the second subset, then tree algorithm would retransmit this subset with a guaranteed collision Information Networks – p.10/24

Tree Algorithms, improvement 1 An improvement would be to upon detecting an idle slot following a collision splitting this second set into two sets and just retransmit the first one Similarly if an idle occurs again, the second subset is again split before retransmission, and so forth The improvement can be stated in the stack case too, each node much keep an additional binary state that is 1 if, for some i ≥ 1 , the last i slots contained a collision followed by i − 1 idle slots, otherwise the state is 0 If feedback is 0 and state is 1, then state stays at 1 and the set on top of stack is split into two that are pushed on stack instead of the old head element Information Networks – p.11/24

Tree Algorithms, improvement 1 The maximal throughput with this improvement is 0 . 46 packets per slot In practice there is a problem with this improved algorithm, if an idle slot is incorrectly perceived by the receiver as a collision then the algorithm continues splitting indefinitely Thus, in practice, after some number h of idle slots following a collision, the algorithm should be modified to not split any further but instead retransmit the current subset If the feedback is very reliable h can be fairly large, otherwise h should be small Information Networks – p.12/24

Tree Algorithms, improvement 2 Consider a collision followed by another collision. Let x be the number of packets in first collision and x L and x R the number of packets in the L and R subsets from the split, thus x = x L + x R Assuming that x is Poisson and that the splitting into L and R is done randomly so that x L and x R are independent (and due to random splitting Poisson) Given the two collisions, we know x L + x R ≥ 2 and x L ≥ 2 and since the latter implies the former conditioning on both is the same as conditioning on the latter only Information Networks – p.13/24

Tree Algorithms, improvement 2 Given the feedback of two collisions we thus get P ( x R = k | x L ≥ 2) = P ( x R = k ) since x L and x R are independent On the other hand P ( x L = k | x L ≥ 2) = P ( x L = k ) /P ( x L ≥ 2) if k ≥ 2 (and 0 otherwise) Thus most packets are probably in first slot, and the small expected number of packets in R makes it reasonable to not devote a slot to R but instead count the packets in this set as waiting new arrivals With this improvement and splitting done on arrival time we get the first-come first-serve (FCFS) splitting algorithm Information Networks – p.14/24

Variants of the tree algorithm Tree algorithm requires all nodes to monitor channel feedback and keep track of when each CRP ends If new arrivals instead just join the subset of nodes at head of stack, and only backlogged nodes monitor the feedback, we get an algorithm called unblocked stack algorithm In contrast the tree algorithm is often called blocked stack algorithm, since new arrivals are blocked until the end of current CRP Since new arrivals are added to head of stack, collisions involve a somewhat larger number of packets on the average Information Networks – p.15/24

Variants of the tree algorithm Because of likelihood of three or more packets in collision higher maximum throughput can be obtained by splitting into three subsets rather than two Maximal throughput for unblocked stack algorithm is 0 . 40 Information Networks – p.16/24