1 Introduction FRAME LOSSES DUE TO BUFFER OVERFLOWS IN FAST PACKET NETWORKS. Abstract CH2826-5/90/0000/0132/$01.OO IEEE 132

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1 FRAME LOSSES DUE TO BUFFER OVERFLOWS IN FAST PACKET NETWORKS AMIT BHARGAVA AND MICHAEL 6. HLUCHYJ Codex Corporation 20 Cabot Boulevard Mansfield, MA Abstract When a fast packet subnetwork is used to transport higher layer Protocol Data Units (PDU), each PDU is typically segmented into multiple fast packets which are then individually relayed along a fixed path through the fast packet subnetwork. At the far edge of the subnetwork, the PDU is reassembled and checked for missing or corrupted data. The loss of one or more of its constituent fast packets causes the entire PDU to be discarded at the subnetwork edge, with retransmissions initiated by higher layers in the protocol stack. An analytical method is derived to estimate the PDU frame loss probability as a function of buffer size when identical sources are multiplexed on a fast packet link. The generation of PDUs from each source is govemed by an On-Off source model, and upper and lower bounds on the frame loss probability are derived. The analysis takes into account the correlated nature of fast packet losses within a PDU due to buffer overflows. As one would expect, the required buffer size for a given frame loss probability is directly proportional to the expected PDU frame length, and the loss probability decreases exponentially with increasing buffer size. 1 Introduction A key technology trend for both private and public communication networks is the movement toward fast packet techniques in the multiplexing and switching of a wide diversity of traffic types. In a fast packet network, all traffic is transported in the form of packets that are relayed from node to node along fixed paths using address information in the packet header. Unlike traditional packet networks, link-by-link flow control and error recovery are not done within a fast packet subnetwork. The intermediate nodal processing of fast packets is limited to the simple functions of fast packet framing, error checking, address look-up/translation, switching and queueing. If a fast packet header is found in error or there are insufficient buffers available at an intermediate node, the entire fast packet is discarded. Where required, retransmissions are initiated at the subnetwork edge or in some cases are left to the end systems. The application of fast packet techniques for framed data is best described in the context of a reference model. As Figure 1 illustrates, fast packets are incorporated in the model below the Data Link Control (DLC) of Layer 2. In a traditional packet network, a Layer 2 (e.g., LAPB) Protocol Data Unit VDU) is framed (delimited by flags and bit stuffed) and passed over a physical channel to the next node. In a fast packet network, the Layer 2 PDU is generally segmented into smaller units (i.e., fast packets) each containing an address identifying the specific Layer 2 entity to which it belongs. This function is provided by the Fast Packet Adaption (FPA) layer shown in Figure 1. Fast packets are multiplexed (for those Layer 2 entities that share a common physical channel to another node) by the Fast Packet Relay (FPR) layer before transmission over the physical channel. After being relayed though one or more intermediate nodes, the pieces of the original Layer 2 PDU that were placed in separate fast packets arrive at the edge node, are reassembled, and the entire PDU is checked for errors (i.e., random bit errors and missing bytes), with error recovery generally done by the Layer 2 DLC or left to higher layers. In some cases the DLC is terminated by the end system (i.e., the Layer 2 DLC lies outside the subnetwork), and the subnetwork provides a simple frame relay service. Here the frame relay function in the edge node is done in the RA layer. As before, the received Layer 2 PDU from the end system is divided into fast packets that are relayed through the fast packet subnetwork until they reach the destination edge node. At the destination cdgc node, the Layer 2 PDU is reassembled from the received fast packets and checked for errors. If an error is detected, the PDU is discarded, otherwise it is passed on to the end system. In some CH2826-5/90/0000/0132/$01.OO IEEE 132

2 PH DL FPR FPA DLC Figure 1 : Fast Packet Layered Model cases, the FF'A and FPR functions are also included in the end system. Note that the service provided to the Layer 2 DLC, whether at an edge node or in the end system, is one in which either a Layer 2 PDU frame is delivered without errors and without missing data or is not delivered at all. Hence, one performance measure of interest to the Layer 2 DLC is the (PDU) frame loss rate. It is the frame loss rate along with the data rate and edge-to-edge (or end-to-end) delay that will determine the selection of an appropriate retransmission strategy (e.g., go-back-n or selective repeat [2]). In this paper we examine the frame loss performance due to buffer overflows in a fast packet network. At the Fast Packet Relay layer, fast packets from different Layer 2 entities (or sources) are statistically multiplexed onto a single link, with a queue to absorb fluctuations in packet amvals. Though the sum total of the average rates of these sources can be kept less than the link capacity (B), the instantaneous rate can exceed the link capacity causing the buffer queue to build and possibly overflow. The model that we use to analyze the behavior of the queue is termed the On-Off model since the sources are either in the on state generating bits or in the off state when they are silent. This model assumes that when the sources are on, they generate bits that ff ow out in a continuous manner at a rate V. If the link capacity is larger than the total rate at which bits enter, the queue does not grow. However, if the number of sources that are on at any instant is larger than c = B/V, the queue grows and may eventually overflow. In this paper the interval of time for which the queue is held at its limit is termed the loss period, which ends as soon as the number of on sources falls below c. Note that this model is approximate in that it ignores the effects of packetization and the highfrequency fluctuations in the queue length that occur when packets arrive together in a short interval of time (even though the number of sources that are on may be less than c). On-Off modells have been studied in the past with both infinite and finite queues in References [l] and [4], respectively. We will use results derived in these papers in our analysis. The model assumes that the sources come on and go off for random intervals of time that are exponentially distributed with rates l/t'o~ and ~/TQFF, respectively, where TON is the average on time and TOFF is the average off time for a source. The number of bits that a source generates in one on time (If x TQN on the average), corresponds to one frame of data, which we assume is also the Layer 2 PDU in this analysis. A total of N sources are multiplexed on the link and are assumed to be identical. Section 2 describes the analysis that is used to calculate the bounds on the frame loss probability. In Section 3, we present numerical results for typical parameters, verification of the analysis by simulation, and a comparison of the analysis with simpler models. Finally, Section 4 presents the conclusions. 133

3 Time Figure 2: Fluctuations in the Number of On Sources and the Queue Length with Time 2 Analysis for the Frame Loss Probability Figure 2 illustrates the fluctuations in the number of on sources with time, and the consequent changes in queue length. The periods during which the number of on sources exceeds c is termed an overload period and the interval of time during which the queue is held at its limit is called a loss period. Note that not every overload period results in losses. Focusing our attention on an arbitrary interval of time t, we define F(t) = Number of frames lost in time t A(t) = Number of frames amved in time t. Hence, we may write assuming the limits exist. Let Then, n(t) = fi = Number of loss periods in time t Number of frames lost in the ith loss period in time t. (1) where E[f] is the average number of frames lost in a loss period. Also, Therefore, (3) We first find the average number of frames lost in a loss period, E[f]. When a loss period starts, some of the sources that are on and are sending frames will lose fast packets. The worst case is that all of the sources that are on lose packets at this instant. In the best case, since the link can accommodate c sources, only (i - LcJ) sources would lose packets, where i is the number of sources that are on at the instant of the start of the loss period and IC] indicates the largest integer less than or equal to c. In reality, the number of sources that are affected would be between these two limiting values which yield an upper and a lower bound, respectively. In addition to the frames lost at the start of the loss period, each new source that comes on during the loss period will cause the dropping of an additional frame since additional sources cannot be accommodated on the link. The finite queue is held at its limit for the duration of the loss period. Therefore, where E[f ]i ] is the expected number of frames lost in a loss period, given i sources are on at the beginning I34

4 of the loss period, and v; is the probability of having i sources on at the start of the loss period. Defining E [ U 1 i ] to be the average number of sources that come on during the loss period (until the loss period ends, when the number of on sources falls below c), given that i were on at its start, we have < E[Uli]+i (upper bound) E[U )i ] + i - LcJ (lower bound) (6) To find vi, we draw upon the analyses in [ 11 and [4] and approximate it by vi = (7) where U; is the joint probability that i sources are on and the queue is held at its limit (i.e., the probablity of having i sources on during the loss period rather than at the start of the loss period). The 21; s can be found for a finite queue or an infinite queue using the methods described in References [4] and [ 11, respectively. To find E[ U Ii ] we use the birth-death process model for the number of on sources shown in Figure 3 (A = I/TOFF and p = l/to~). Note that for this model, a loss period can only be entered from a state i where i > Lc]. The loss period ends when the transition from state IC] + 1 is made to state LcJ. The average number of sources that come on during this interval is equal to the average number of up transitions that are made during the loss period (after entering it from state i). 21; ~sj=icj +I uj Defining E[D Ii ] to be the average number of down transitions during the loss period and T~,FJ to be the average number of transitions from state 1 to state IC], we have, E[D(i 3 = E[U li ]+(i- LCJ) (8) T~,L.] = E[U )i ] + E[D li 3 (9) Note that ri,j is the first passage time from state i to state j for a discrete time birth-death process obtained by embedding the continuous time birth-death process at the instants of transition. Using the method outlined in [3], we have where rj,+ 1 can be computed recursively for 0 < j < N from The transition probabilities pj and qj are given by The next quantity that is needed for calculating the loss probability, 9, is Lz T. We first define L; = L ena of the ith loss period. Noting that the fraction of time spent in loss periods in an interval of length t converges to the probability of the queue being held at its limit, as t + m, we have Multiplying and dividing the left hand side of this equation by n(t) and then rearranging terms we obtain, Solving for E[U ]i ] and substituting in Equations 6 and 5, we obtain The analysis for frame loss probabilities in this paper is done for a finite queue. The infinite queue analysis of [ 11 can be used to approximate the 11, s since it does not require the solution of large systems of linear equations like [4] and is therefore easier to use for computations. where E[L] is the average length of a loss period. Therefore, Once again, we solve for E[L] in terms of the first passage times from state?: to Lc], T;,J~J, for the continuous 135

5 Figure 3: Birth-death process for the number of on sources C time birth-death process, N Using the approach outlined in [3] and defining Fj = Tj,j- 1 we have i ~i,~cl = rj (20) where for 0 < j < N j=lcj+l 3 Numerical Results and Discussion ig L4 9% b 2! I Buffer Size (in Frames) Figure 4: Comparison of bounds with simulation results - UB...IJ e PEL. FL=S Sim. Sim. FLdOO In this section we present some numerical results based on the analysis in Section 2. Figure 4 compares the lower bound (LB) and the upper bound (UB) with loss probabilities obtained by simulation, for average frame lengths (FL) of 1, 50 and 100 fast packets, respectively, but keeping the ratio A/p constant. These curves are generated for N = 8,c = 4 and a link utilization (p) of 0.8. Note that the buffer sizes are in units offrames; in these normalized units the frame loss probability predicted by the analysis is exactly the same for all frame sizes. The dependence of 0 on the ratio X/p (rather than X and ji individually) can be verified algebraically and is expected intuitively, since changing the quantities but keeping their ratio constant is like changing the unit of time which does not alter F(t)/A(t) as t --t 00. The size of the buffer in bytes grows in direct proportion to the expected frame length. The curves are plotted for loss probabilities that would be required in in a typical network, given that the overall error performance would still be bounded by the bit error rate on the links. The simulation takes into the account the packetization of the frames and the fact that fast packets amve to the queue, rather than the bits flowing in smoothly. That is, the simulation includes both the high-frequency queue fluctuations caused by the arrival pattem of the packets from sources in the on state, and the low-frequency fluctuation caused by the smooth flow of bits entering the queue at rate iv, i = 0, 1,..., N. To maintain the desired utilization on the link for a fair comparison with the analysis, we assume that the multiplexer can accept fractions of packets when it is about to overflow. For example, if 16 bytes of buffer are left when a 64-byte packet amves, 16 bytes flow into the buffer and 48 bytes are dropped. The exponential length of the on time is accounted for by generating shorter packeb at the end of the on time. Notice that the lower and upper bounds are parallel to 136

6 each other on a logarithmic scale. This can be expected if the ratio of the two bounds is roughly a constant for different values of the buffer size. From the results of Section 2, it is easy to show - E[flupper -- =1+ E [fllower E [f llower (23) where the subscripts lower and upper refer to the lower and upper bound analyses, respectively. Our observation can be explained if we can show that the quantity E[f]lower does not change significantly with the buffer size. Note that E[f]lower consists of two components: the number of frames that are lost at the start of the loss period and the number that are lost during the loss period. The latter depends (in our analysis) only on the on-off behaviour of the sources, which is independent of the buffer size. The former quantity, however, is not strictly independent of the buffer size (for example, for a buffer of size zero, it is always one frame), but we conjecture that as more buffer is added the distribution of this quantity rapidly approaches the conditional distribution of the number of on sources i, given i > IC]. Intutively, this could be expected since by increasing the buffer, we are giving the sources more time in the overload states (i.e., the number of on sources being larger than LcJ) without the buffer overflowing. Therefore, at the instant when the queue hits its maximum, the number of sources that are on would roughly be some number picked from the distribution of the number of on sources (conditioned on being in the overload states), and therefore independent of the buffer size. What changes with the buffer size is the probability of getting into a loss period, and causes both curves to slope down. The second observation from Figure 4 is that the simulation points lie between the two bounds though the points predict a consistently larger loss than the lower bound analysis. The lower bound analysis underestimates the loss due to the fact that packets from various sources can line up and cause losses when the queue is close to its limit, even though the total number of on sources is less than LcJ. This?he apparant increase in deviation from the lower bound analysis curve for larger buffer sizes is due to the limitations of computing resources in the simulation. For large frames and large buffers the simulation must be run for unreasonable lengths of time to obtain more accurate values. corresponds to a high-frequency component of the queue fluctuation. The upper bound analysis, however, still predicts a larger loss as it always assumes that i frames are lost at the start of a loss period if i sources are on. In addition, if the queue length hovers around its limit, multiple loss periods can occur during the duration of a single frame. Even though the effect of dropping packets from that frame in different loss periods is losing just one frame, the upper bound analysis counts a lost frame for each loss period. The conclusion of these observations is that in the range of interest, the frame loss probability lies between the two bounds, and we use these to characterize the system behavior further. U i2 i5 Buffer Size (in Frames) Figure 5: Comparison of the bounds with analyses assuming independent packet losses Figure 5 compares the bounds with a simplified frame loss analysis (marked SA on the figure) for frame lengths of 1, 50 and 100. In this analysis we assume that packets are lost independently of each other with a packet loss probability 8. In this case the frame loss probability is given by (see appendix for details) In( 1-6) QPapprox = In(1-6) - ft where M is the average frame length in packets. As can be seen from Figure 5, the analysis predicts a frame loss probability that is proportional to the average frame length and can predict either a larger or a smaller result for large and small frame lengths, respectively. For a frame length of 1 packet we expect a frame loss Probability prediction that is close to the packet loss probability, since for each packet dropped, only 1 frame 137

7 is dropped on the average. In this paper, however, 8 is computed using the method described in [4], which is based on counting the fraction of information lost. The actual packet loss for an average frame length of 1, will be larger since the loss of information equal to a fraction of a packet results in the whole packet being dropped. Consequently, the curve for the independent packet loss with a frame length of 1 lies below the lower bound curve, instead of close to it. For larger frame lengths, however, this error decreases. In addition, as the frame length is increased, the simplified analysis assumes that packets are dropped independently from all frames rather than multiple packets being dropped from the same frame. Naturally this predicts a higher frame loss rate and is unsuitable for estimating frame loss probabilities Buffer Size (in Frames) Figure 7: Effect of increasing N and TOFF with fixed p, TON and c I I I I UB 1 - N=8 fixed p), the sources look like continuous bit streams and since N must now be less than or equal to c, the model will predict no loss at all. This explains why a worse performance is seen for larger N. Y i2 Buffer Size (in Frames) Figure 6: Effect of increasing N and c in proportion for fixed p, Torv and TOFF In Figure 6, we examine the effect of varying N and c in proportion while keeping the link utilization (p) and the on and off times of the source constant. Notice that the bounds move lower as we increase AT since the law of large numbers decreases the probability of getting into the loss periods. The bounds also move further apart for larger N due to the factor IC] in Equation 23. Finally, Figure 7 shows the effect of changing N and TOFF with fixed TON, c and p. A larger N now implies that the aggregate stream entering the multiplexer looks more 'Poisson-like' and in the limit as N -, 0;) the interamval times of the frames will be exponentially distributed (though the bits will still flow in at the rate 17). On the other hand, if TOFF is made zero for all sources to get the smallest N (with 4 Conclusions We have presented an analytical method for estimating bounds on the frame loss probabilities as a function of buffer size in fast packet networks. The analysis was validated by simulation and shown to be accurate in the region of practical interest. Since edge-to-edge protocols retransmit frames in a fast packet network, we noted that the frame loss probability is the more important quantity in establishing the quality of service of the fast packet subnetwork for framed data. We showed that using a simple analysis where packets are assumed to be dropped independently yields inaccurate results and it is important to use an analysis (such as the one in this paper) that accounts for the correlation of packet losses in a frame. The analysis also indicates, as one would expect, that the amount of buffering (in bytes) required to attain a given frame loss probability is proportional to the average size of the frame. 138

8 APPENDIX Frame loss probability based OII independent packet losses In this approximation, we assume that a frame of average length M packets is lost due to the loss of one of its packets, each of which is lost independently with a probability 8. Let X be the length of an individual frame in packets and since the on time is exponentially distributed the density function of X is The approximate frame loss is given by: 03 cppapprox = 1 Pr[Frame is lost 12 ]fx.(z)da: (A.2) The frame is not lost only if none of its packets lost. Therefore are where r.1 to 2. is the smallest integer greater than or equal References [ 11 D. Anick, D. Mitra, and M. M. Sondhi, Stochastic theory of a data-handling system with multiple sources, Bell System Technical Journal, vol. 61, 1982, pp [2] D. Bertsekas and R. Gallager, Data Nemorks, Prentice Hall, [3] D. P. Heyman and M. J. Sobel, Stochastic Models in Operations Research, Volume I, McGraw Hill, [4] R. C. F. Tucker, Accurate method for analysis of a packet-speech multiplexer with limited delay, IEEE Transactions on Communications, vol. 36, no. 4, April 1988, pp