TSIN01 Information Networks Lecture 9

Transcription

1 TSIN01 Information Networks Lecture 9 Danyo Danev Division of Communication Systems Department of Electrical Engineering Linköping University, Sweden September 26 th, 2017 Danyo Danev TSIN01 Information Networks Lecture 9 1 / 38

2 Lecture overview Agenda CSMA Slotted ALOHA (contd) CSMA Slotted ALOHA: Pseudo-Bayesian stabilization CSMA Unslotted ALOHA FCFS with CSMA Multiaccess reservations CSMA/CD (Collision Detection): slotted and unslotted Danyo Danev TSIN01 Information Networks Lecture 9 2 / 38

3 Nonpersistent CSMA slotted ALOHA The drift in state n is as before the expected number of arrivals less expected numbers of departures ( D n = λ(β +1 e λβ (1 q r ) n ) λβ + q ) rn e λβ (1 q r ) n 1 q r For small q r we make the approximation (1 q r ) n 1 (1 q r ) n e qrn and get D n λ(β + 1 e g(n) ) g(n)e g(n) where g(n) = λβ + q r n is expected number of attempted transmissions Danyo Danev TSIN01 Information Networks Lecture 9 3 / 38

4 Nonpersistent CSMA slotted ALOHA The drift is negative if λ < g(n)e g(n) β + 1 e g(n) where the numerator is the expected number of departures per state transition and the denominator is the expected duration of a state transition, so the ratio can be interpreted as the departure rate in state n We can plot the departure rate as function of attempted rate as before, for small β this function has a maximum of approximately 1/(1 + 2β) for g = 2β Danyo Danev TSIN01 Information Networks Lecture 9 4 / 38

5 Nonpersistent CSMA slotted ALOHA Danyo Danev TSIN01 Information Networks Lecture 9 5 / 38

6 Nonpersistent CSMA slotted ALOHA Same stability problem as in ordinary slotted ALOHA For fixed q r, g(n) grows with n and when n becomes too large, departure rate is less than arrival rate (drift is positive), leading to yet larger backlogs Expected idle time that a backlogged node must wait before attempting retransmission is β(q r + 2q r (1 q r ) + 3q r (1 q r ) ) = β/q r For small β and modest λ, q r can be quite small without causing appreciable delay This means that the backlog must be very large before instability sets in and the problem is less serious than for ordinary ALOHA Danyo Danev TSIN01 Information Networks Lecture 9 6 / 38

7 CSMA slotted ALOHA stabilization P-persistent CSMA, in which packets are transmitted after idle slots with probability p if they are new arrivals and with some much smaller probability q r if they have had collisions will give a little extra protection against instability A more fundamental way to achieve stability is to do a pseudo-bayesian stabilization as for the ordinary slotted ALOHA All packets are considered backlogged immediately after entering the system At end of each idle slot, each backlogged packet is transmitted with probability q r (ˆn) which will vary with the estimated channel backlog ˆn Danyo Danev TSIN01 Information Networks Lecture 9 7 / 38

8 Pseudo-Bayesian stabilization In state n, expected number of packets transmitted at end of idle slot is g(n) = nq r, packet departure rate is maximized (for small β and q r ) when g(n) = 2β so we choose { 2β q r (ˆn) = min ˆn, } 2β Backlog estimate is updated according to ˆn k (1 q r (ˆn k )) + λβ, ˆn k+1 = ˆn k (1 q r (ˆn k )) + λ(1 + β), ˆn k λ(1 + β), for idle for success for collision Danyo Danev TSIN01 Information Networks Lecture 9 8 / 38

9 Pseudo-Bayesian stabilization Again the update rule for this Pseudo-Bayesian stabilization can be motivated by showing that for an a priori Poisson distribution of n k with mean ˆn k, the a posteriori distribution of n k is Poisson with mean ˆn k (1 q r (ˆn k )) given an idle slot Poisson with mean 1 + ˆn k (1 q r (ˆn k )) given a successful transmission approximately Poisson with mean ˆn k + 2 given a collision Adding the expected arrivals in the three cases yields the suggested update rule Danyo Danev TSIN01 Information Networks Lecture 9 9 / 38

10 Pseudo-Bayesian stabilization When n k and ˆn k are small, then q r is relatively large and new arrivals are scarcely delayed at all When ˆn k n k and n k is large, the departure rate is approximately 1/(1 + 2β) So, for λ < 1/(1 + 2β) the departure rate exceeds the arrival rate (i.e. the drift is negative) and the backlog decreases on average If n k ˆn k is large the expected change in backlog can be positive, but the expected change in n k ˆn k is negative so eventually ˆn k will be close to n k and backlog will decrease; similar to pseudo-bayesian stabilization of ordinary slotted ALOHA Danyo Danev TSIN01 Information Networks Lecture 9 10 / 38

11 Delay of pseudo-bayesian stabilization We can do a similar analysis of the expected queueing delay as for pseudo-bayesian stabilization of ordinary slotted ALOHA Let W i be the delay from arrival of ith packet until beginning of ith successful transmission Average of W i over all i is the expected queueing delay W Let n i be the number of backlogged packets at the instant before packet i s arrival, not counting any packet currently in successful transmission Danyo Danev TSIN01 Information Networks Lecture 9 11 / 38

12 Delay of pseudo-bayesian stabilization n i W i = R i + t j + y i j=1 R i is the residual time until next state transition, t j is the sequence of subsequent intervals until each of the next n i successful transmissions are completed, and y i is the remaining interval until the ith successful transmission starts The backlog is at least 1 in all of the state transition intervals and we make the simplifying approximation that the number of attempted transmissions in each of these intervals are Poisson with parameter g Danyo Danev TSIN01 Information Networks Lecture 9 12 / 38

13 Delay of pseudo-bayesian stabilization This is different from the analysis of ordinary stabilized slotted ALOHA, where we assumed that a successful transmission always occurred when backlog was 1. This difference is because q r in CSMA is chosen to be small The expected value for each t j is given by E[t] = e g (β+e[t])+ge g (1+β)+[1 (1+g)e g ](1+β+E[t]) The first term corresponds to an idle in the first state transmission interval The second term corresponds to a success, and the third term to a collision Danyo Danev TSIN01 Information Networks Lecture 9 13 / 38

14 Delay of pseudo-bayesian stabilization Solving for E[t] gives E[t] = 1 + β e g ge g This is the reciprocal of expected departure rate and thus is approximately minimized by g = 2β Averaging over i and using Little s theorem we get W = E[R] + λw E[t] + E[y] Danyo Danev TSIN01 Information Networks Lecture 9 14 / 38

15 Delay of pseudo-bayesian stabilization The expected residual time E[R] is approximated by observing that the system spends a fraction λ(1 + β) of the time in successful state transition intervals, and the expected residual time for arrivals in these intervals is (1 + β)/2 For small β, the fraction of time spend in collision intervals is negligible compared with that for success and the residual time for idle intervals is negligible too Thus, E[R] λ(1 + β)2 2 Danyo Danev TSIN01 Information Networks Lecture 9 15 / 38

16 Delay of pseudo-bayesian stabilization Finally E[y] is just E[t] less the time for a successful transmission, E[y] = E[t] (1 + β) Putting all these together we get W λ(1 + β)2 + 2(E[t] (1 + β)) 2(1 λe[t]) This expression is minimized over g by minimizing E[t] which is 1 + 2β at g = 2β (for small β), thus W λ + 2 2β 2(1 λ(1 + 2β)) Danyo Danev TSIN01 Information Networks Lecture 9 16 / 38

17 Delay of pseudo-bayesian stabilization The delay for stabilized CSMA ALOHA W λ + 2 2β 2(1 λ(1 + 2β)) is similar to the M/D/1 queueing delay with service time µ = 1 (we assumed time measured in average packet transmission time) W = λ 2(1 λ) By stabilizing CSMA ALOHA we modify q r with the backlog to maintain a departure rate close to 1/(1 + 2β) whenever a backlog exists Danyo Danev TSIN01 Information Networks Lecture 9 17 / 38

18 Unslotted CSMA ALOHA In slotted CSMA ALOHA we assumed that all nodes were synchronized to start transmissions only at time multiples of β in idle periods We now remove that restriction and assume that when a packet arrives its transmission starts immediately if the channel is sensed to be idle If the channel is sensed to be busy, or if transmission results in a collision, the packet is regarded as backlogged Each backlogged packet repeatedly attempts to retransmit at randomly selected times separated by independent exponentially distributed random waiting times τ with probability density xe xτ Danyo Danev TSIN01 Information Networks Lecture 9 18 / 38

19 Unslotted CSMA ALOHA If the channel is idle at one of these times, the packet is transmitted and this continues until the packet has been successfully transmitted We again assume propagation and detection delay of β, so if one transmission starts at time t, another node will not detect that the channel is busy until time t + β thus causing the possibility of collisions For an idle period that starts with a backlog of n and infinite-node assumption, the time until first transmission starts is exponentially distributed with rate G(n) = λ + nx Note that G(n) is now the attempt rate in packets per unit time, previously g(n) was in packets per idle slot Danyo Danev TSIN01 Information Networks Lecture 9 19 / 38

20 Unslotted CSMA ALOHA After initiation of this first transmission, the backlog is n (if a new arrival started the transmission) or n 1 (if a backlogged packet started) The time from this first transmission until another node senses the channel (to transmit a backlogged or new packet) is exponentially distributed with rate G(n) or G(n 1) Collision occurs if this sensing is done within time β Probability of collision is P r(τ < β) = 1 e βg(n) (or 1 e βg(n 1), but the difference is small if βx is small) Probability of a successful transmission following an idle period is approximately e βg(n) Danyo Danev TSIN01 Information Networks Lecture 9 20 / 38

21 Unslotted CSMA ALOHA The expected time from beginning of one idle period until next is 1/G(n) + (1 + β) where 1/G(n) is the time until first transmission and (1 + β) is time until first transmission ends and the channel is detected as idle again If a collision occurs there is a slight additional time (< β) until the packets causing the collision are not detected, this time is however negligible since already β is negligible The departure rate when backlog is n is then e βg(n) 1/G(n) + (1 + β) Danyo Danev TSIN01 Information Networks Lecture 9 21 / 38

22 Unslotted CSMA ALOHA For small β, the maximum departure rate is approximately 1/(1 + 2 β), occurring when G(n) 1/ β It is slightly lower than for the slotted case The reason is the same as when CSMA is not used: for a given attempt rate, collisions are more likely in an unslotted system For CSMA with small β, the difference is quite small and in a slotted system β has to be larger due to synchronization inaccuracies and worst-case propagation delay Thus unslotted ALOHA is the natural choice for CSMA Danyo Danev TSIN01 Information Networks Lecture 9 22 / 38

23 Unslotted CSMA ALOHA CSMA unslotted ALOHA has the same stability problems as all the ALOHA systems It can be stabilized with a pseudo-bayesian stabilization strategy similar to the CSMA slotted ALOHA Danyo Danev TSIN01 Information Networks Lecture 9 23 / 38

24 FCFS splitting algorithm with CSMA Relatively little can be gained (in throughput or delay) by using splitting algorithms with CSMA For small β, the FCFS splitting algorithm has the same maximum throughput as slotted ALOHA This is not surprising when realizing that without CSMA the major advantage of FCFS algorithm is its efficiency in resolving collisions, and with CSMA collisions rarely occur When collisions do occur they are resolved in both strategies by retransmission with small probability Danyo Danev TSIN01 Information Networks Lecture 9 24 / 38

25 Multiaccess reservations If the data packets are long, it is inefficient to waste long slot times sending nothing or sending colliding packets It would be far more efficient to send very short packets (either in contention mode or in TDM mode) to reserve longer noncontending slots for the actual data In this way, the slots wasted by idles or collisions are all short, leading to a higher overall efficiency Assume that each reservation packet requires for transmission v time units, which is much less than the one time unit needed for a data packet Danyo Danev TSIN01 Information Networks Lecture 9 25 / 38

26 Multiaccess reservations The format of the reservation packet is unimportant, it simply has to contain enough information to establish the reservation With a (0, 1, e) instantaneous feedback, the mere existence of the reservation packet is enough After a successful transmission of a reservation packet, the next full time unit is automatically allocated for transmission of the corresponding data packet The reservation packets can use any multiaccess strategy, e.g., TDM, slotted ALOHA, or splitting algorithms Danyo Danev TSIN01 Information Networks Lecture 9 26 / 38

27 Multiaccess reservations Let S r be the maximum throughput, in successful reservation packets per reservation slot, of the algorithm used for reservation packets (1 for TDM, 1/e for slotted ALOHA, for splitting) Over a large number of reservations, the time required per reservation approaches v/s r Then, the total time per data packet approaches 1 + v/s r The maximum throughput S in data packets per unit time is 1 S = 1 + v/s r If v is small (e.g. 0.01), then S 1 irrespective of the (throughput S r of the) reservation strategy Danyo Danev TSIN01 Information Networks Lecture 9 27 / 38

28 CSMA/CD Ethernet is a widely used technique for local area networks, a number of nodes are all connected onto a common cable so that when one node transmits a packet (and all others are silent), all other nodes hear that packet In addition, as in carrier sensing, a node can listen to the cable before transmitting Finally because of the physical properties of the cable, it is possible for a node to listen to the cable while transmitting Thus, if two nodes start to transmit almost simultaneously, they will shortly detect a collision in process and both cease transmitting Danyo Danev TSIN01 Information Networks Lecture 9 28 / 38

29 CSMA/CD This technique is called CSMA/Collision Detection If one node starts transmitting and no other node starts before the first node s signal has propagated throughout the cable, the first node is guaranteed to finish its packet without collision Thus, we can view the first portion of a packet as making a reservation for the rest For analytic purposes it is easiest to visualize Ethernet in terms of slots and minislots, the minislots are of duration β which denotes the time for the signal to propagate throughout the cable and be detected Danyo Danev TSIN01 Information Networks Lecture 9 29 / 38

30 Slotted CSMA/CD All nodes are synchronized into minislots of duration β If only one node transmits in a minislot, all other nodes will detect the transmission and not use subsequent minislots until the entire packet completed If more than one node transmits in a minislot, each transmitting node will detect this and cease transmitting by the end of the minislot Thus, the minislots are used in contention mode, and when a successful transmission occurs in a minislot, the channel is reserved for the completion of the packet CSMA/CD can be analyzed with a Markov chain in the same way as CSMA ALOHA Danyo Danev TSIN01 Information Networks Lecture 9 30 / 38

31 Slotted CSMA/CD We assume that each backlogged node transmits after each idle slot with probability q r, and that the number of nodes transmitting after an idle slot is Poisson with parameter g(n) = λβ + nq r We consider state transitions at the end of idle slots: after time β (if no transmission occurs); after time 1 + β (if one transmission occurs); after time 2β (if collision occurs; this is because nodes must hear an idle slot after the collision before transmitting); Danyo Danev TSIN01 Information Networks Lecture 9 31 / 38

32 Slotted CSMA/CD The expected length of the interval between state transitions is then E[interval] = β + g(n)e g(n) + β(1 (1 + g(n))e g(n) ) The expected number of new arrivals between state transmissions is λ times this interval, so the drift in state n is λe[interval] Psucc, the probability of success is g(n)e g(n), so we get that the drift is negative if λ < g(n)e g(n) β + g(n)e g(n) + β(1 (1 + g(n))e g(n) ) The right-hand side of the inequality can be interpreted as the departure rate in state n Danyo Danev TSIN01 Information Networks Lecture 9 32 / 38

33 Slotted CSMA/CD The departure rate is maximized over g(n) at g(n) = 0.77 and the resulting maximum is 1/( β) CSMA/CD can be stabilized with e.g. the pseudo-bayesian technique and then the maximum λ for which the system is stable is λ = 1/( β) The constant 3.31 is dependent on the detailed assumptions about the system, different values can be obtained by making different assumptions If β is very small, as usual in Ethernet, this value is not very important Danyo Danev TSIN01 Information Networks Lecture 9 33 / 38

34 Unslotted CSMA/CD Unslotted CSMA/CD makes more sense than slotted, both because of the difficulty of synchronizing on short minislots and the advantages of capitalizing on shorter than maximum propagation delays when possible The exact analysis of unslotted CSMA/CD is somewhat messy and complicated, e.g. nodes closer together on the cable detect collisions faster than those more spread apart A conservative bound on throughput can be obtained by finding bounds on all relevant parameters from the end of one transmission to the end of next Assume that each node initiates transmissions according to independent Poisson processes whenever it senses the channel is idle, assume G is the overall Poisson intensity Danyo Danev TSIN01 Information Networks Lecture 9 34 / 38

35 Unslotted CSMA/CD All nodes sense beginning of idle period at most β after end of transmission, expected time to beginning of next transmission is an additional 1/G This next packet will collide with some later starting packet with probability at most 1 e βg and the colliding packets will cease transmission after at most 2β The packet will be successful with probability at least e βg and will occupy 1 time unit Departure rate is success probability divided by expected time of a success or collision; so S > e βg β + 1/G + 2β(1 e βg ) + e βg Danyo Danev TSIN01 Information Networks Lecture 9 35 / 38

36 Unslotted CSMA/CD This departure rate will be maximized at βg = 0.43 and the maximum value is 1/( β) The analysis is very conservative, but if β is small throughput close to 1 can be achieved and the difference compared to the result for slotted CSMA/CD is not large Maximum stable throughput approaches 1 with decreasing β as a constant times β for CSMA/CD, whereas the approach is as a constant times β for CSMA, the reason is that collisions are not very costly with CSMA/CD and thus a higher attempt rate can be used Danyo Danev TSIN01 Information Networks Lecture 9 36 / 38

37 Unslotted CSMA/CD CSMA/CD (and CSMA) becomes increasingly inefficient with increasing bus length, increasing data rate, and decreasing packet size Recall that β is in units of data packet duration, thus if τ is propagation delay and detection time in seconds, C is raw data rate on the bus, and L is average packet length, then β = τc/l Neither CSMA nor CSMA/CD are reasonable system choices if β is more than a few tenths Danyo Danev TSIN01 Information Networks Lecture 9 37 / 38

38 Next lecture Topics Packet radio networks (PRN) TDM and FDM for PRN Slotted ALOHA for PRN Routing Broadcasting: flooding Danyo Danev TSIN01 Information Networks Lecture 9 38 / 38