I 2 (t) R (t) R 1 (t) = R 0 (t) B 1 (t) R 2 (t) B b (t) = N f. C? I 1 (t) R b (t) N b. Acknowledgements

Transcription

1 Proc. 34th Allerton Conf. on Comm., Cont., & Comp., Monticello, IL, Oct., Service Guarantees for Window Flow Control 1 R. L. Cruz C. M. Okino Department of Electrical & Computer Engineering University of California, San Diego La Jolla, CA Abstract After presenting some new insight into the concept of \service curves," we derive a \service curve guarantee" for a window ow control protocol with cross-trac characterized by burstiness constraints. Our approach is convenient for studying the end-to-end behavior of hop-by-hop window ow control, and has an interesting relationship with a linear feedback system under the \min-plus" algebra. For ane burstiness constraints on the cross-trac, we nd that a window size proportional to the sum of the burstiness parameters of the cross-trac and the user bandwidth delay product is sucient to maximize guaranteed throughput. In addition, we nd that buers need not be as large as window sizes for lossless operation with large propagation delays. 1 System Model We will consider the two server system depicted in Figure 1, which models a window ow protocol. Trac from a source is generated according to a function of time R, called a rate function, such that R (t) is the instantaneous rate at which trac is being generated at time t. The trac from the source feeds a buer, called the rst buer. Trac departs the rst buer according to the rate function R 1. We assume that the system is empty at time. Let B 1 (t) denote the amount of trac held in the rst buer at time t. Thus B 1 (t) = R ()d? R 1 ()d : (1) A server, called the rst server, governs the rate R 1 at which trac departs the rst buer. The server has a transmission capacity of C bits/sec, so that R 1 (t) C for all t. In fact, the rst server handles other sources of trac, called \cross-trac," at time t at rate I 1 (t), where I 1 (t) C; thus R 1 (t) C? I 1 (t). Trac departs the rst buer at rate R 1 (t) and feeds a \network element", N f, which serves trac in a FIFO manner at rate R f (t). Recalling that the system is empty at time, the amount of trac held in N f at time t is thus B f (t) = R 1 ()d? R f ()d : () Network element N f feeds another buer, called the second buer. Trac departs the second buer according to the rate function R (t). The amount of trac held in the 1 This work was supported by the National Science Foundation under grants NCR and NCR , by the Army Research Oce under grant FRI DAAH , and by the Air Force Oce of Scientic Research under grant F

2 I 1 (t) I (t) R (t) B 1 (t) R 1 (t) C R 1 (t) N f f R (t) B (t) R (t) C R (t) I 1 (t) Acknowledgements I (t) R b (t) N b Figure 1: Two Servers in Tandem. second buer at time t is thus B (t) = R f ()d? R ()d : (3) A second server governs the rate R at which trac departs from the second buer. The second server also has a capacity of C bits/sec, and serves cross-trac from other sources at rate I (t) at time t, where I (t) C; thus R (t) C? I (t). Trac departing the rst buer is subject to \window ow control," whereby trac departing the rst buer may have to wait for acknowledgements from the the second server. More specically, as trac from the original source departs the second buer, acknowledgements are correspondingly generated by the second server and sent back to the rst server via a network element N b. The rate at which acknowledgements are generated at the second server at time t is R (t). The network element N b operates in a FIFO manner and serves acknowledgements at rate R b (t). Thus, the amount of acknowledgements in N b at time t is B b (t) = R ()d? R b ()d : (4) The window ow control protocol operates with respect to a positive parameter K, called the \window size." The rst server must insure that no more than K units of trac are unacknowledged. Thus, trac in the rst buer may have to wait for acknowledgements to arrive before being eligible for service at the rst server. The total amount of trac unacknowledged (sometimes known as the number of outstanding credits or tokens) at time t is denoted as T (t). Using the denitions above, it follows that T (t) = B f (t) + B (t) + B b (t) = R 1 ()d? R b ()d : (5) The rst server serves trac from the rst buer as fast as possible, but insures that T (t) K for all t. More specically, R 1 (t) = 8 >< >: C? I 1 (t), if B 1 (t) > and T (t) < K minfc? I 1 (t); R (t)g, if B 1 (t) = and T (t) < K minfc? I 1 (t); R b (t)g, if B 1 (t) > and T (t) = K minfc? I 1 (t); R (t); R b (t)g, if B 1 (t) = and T (t) = K : (6)

3 Similarly, the second server serves trac from the second buer as fast as possible: R (t) = ( C? I (t), if B (t) > minfc? I (t); R f (t)g, if B (t) = : The system evolution is completely determined by (1)-(7) and the functions R, I 1, I, R f, and R b. We characterize the network elements N f and N b by \service curves" S f and S b (dened below), respectively. Finally, we assume general \burstiness constraints" [1] on the cross-trac, i.e. we assume that there exist non-negative non-decreasing functions b cross i such that for i = 1; and all s t there holds 1.1 Discussion s (7) I i ()d b cross i (t? s) : (8) The model described above has a number of applications. For example, it may model a single hop within an ATM network that uses credit based ow control. The network element N f would model forward propagation delay in this case. It has been proposed that acknowledgements only be sent back periodically, rather than continuously, and so the network element N b could model a combination of backward propagation delay and jitter caused by accumulation of acknowledgements until a burst of acknowledgements is sent back. Another possible application is the situation in which end-to-end window ow control is applied across multiple hops. In this case, N f would model a combination of queueing and propagation delay suered in the forward path to the destination. Similarly, N b would model a combination of queueing and propagation delay in the backward path for acknowledgements returning to the rst server. Alternatively, the model could describe the ow of data across a protocol layer within a host, whereby processes that feed a buer enter a blocked state when the buer reaches capacity, and the processes provide bursty service due to multi-tasking within an operating system. More generally, the model is of interest in manufacturing networks where window ow control is induced naturally by limited storage space for parts being serviced by a sequence of machines, e.g. see [8]. The model described above is a generalization of a model previously proposed in [3], where N f and N b were not present (i.e. R f R 1 and R b R ), b cross i (x) = i + i x (ane burstiness constraints on cross-trac), and B 1 () = 1 (innite supply of packets at the source). For this special case, a lower bound on the throughput of the system, lim inf t!1 (1=t) R t R, was derived using a \Lyapunov function approach." Although this approach yielded tight results for two servers in tandem, it is dicult to generalize for more than two servers in tandem (see Figure 3) since nding an appropriate Lyapunov function is problematic. Instead, in [3], an aggregation and decomposition technique was used to recursively reduce the analysis of several servers in tandem to the two server case. In this paper, we use the concept of \service curves" [] to analyze the system described above. This allows us to easily extend the analysis for more than two servers in tandem, to incorporate \propagation" delays, and to consider general burstiness constraints on the cross-trac. 3

4 Before analyzing the model above, we dene service curves and present some new interpretations relating to linear system theory. In Section 3, we present our main results on the two server system discussed in this section. In Section 4, we apply the results of Section 3 to the case of several servers in tandem. Service Curves Consider a system with entering and exiting trac described by the rate functions R in and R out. The amount of data stored in the system at time t is B(t) = R t (R in ()? R out ())d, where we assume B() =. Suppose S is a given non-negative function. To simplify the notation, assume without loss of generality that S(x) = for all x. Building upon the results in [9], the following denition was essentially proposed in [] { it is adapted here to the continuous time case. Denition 1. (Strict Service Curve Guarantee). A system is said to strictly guarantee the service curve S if for all t, there exists s t R with B(s) = and t s R out()d S(t? s). The adjective \strict" was not used in []. It is added here to dierentiate it from a slightly weaker service guarantee which we now introduce: Denition. (Service Curve Guarantee). A system is said to guarantee the service curve S if for all t, there exists s t such that R t R out()d? R s R in()d S(t? s). Note that if B(s) =, then R s R in()d = R s R out()d, so that a strict service curve guarantee implies a service curve guarantee. The converse is not necessarily true. In passing, we note that [1] a (; ) (\leaky bucket") regulator guarantees the service curve S reg (x) = + x, x >. Given two functions F and G dened on the non-negative reals, dene the convolution (over the \min-plus algebra") of F and G, written F G, as F G(x) = minff (x 1 ) + G(x ) : x 1 ; x ; x 1 + x = xg ; where the minimum is replaced by an inmum if necessary. It is easy to verify that the convolution operation is commutative and associative, and that it distributes over the minimum operation. It is straightforward to verify that the following is in fact equivalent to Denition. Alternative Denition. (Service Curve Guarantee). A system is said to guarantee the service curve S if for all t there holds R t R out()d S(t) R t R in()d. Dene the \impulse function" ^(x) = if x, and ^(x) = +1 if x >. Note that for any function F, F ^(x) = F (x). It is interesting to note then that a service curve is the impulse response of the network element, in some sense, under the min-plus algebra. In [], it is shown that if a system strictly guarantees a service curve, then bounds on delay, buer size, and burstiness of the output trac are easily derived, assuming burstiness constraints on the trac arriving to the system. It turns out that identical bounds hold if the service curve guarantee is not necessarily strict. The proofs are almost identical { see [1] for the discrete time case. For completeness, we include these results below. The virtual delay at time t, D(t), is dened as D(t) = minf : and + R out ()d R in ()dg : 4

5 Theorem A. [] Assume that R t s R in()d b(t? s) for all s t. Suppose a system guarantees a service curve of S (not necessarily strict). Then (a) (Buer Requirements) There holds for all t B(t) max f[b()? S()]+ g : (b) (Bound on Delay) There holds for all t D(t) maxfminf : and b() S( + )gg : (c) (Output Burstiness) For all s t there holds R t s R out()d b out (t? s), where b out (x) = maxfb(x + )? S()g : Theorem B. [] (Convolution Theorem) Consider trac owing through a system consisting of n subsystems in tandem, where the i th subsystem guarantees the service curve S i (not necessarily strict). Then the system as a whole guarantees the service curve S net = S 1 S S n. Given the analogy to linear ltering theory, it is natural to ask if there is a concept of a transform domain. Given a function F dened on the non-negative reals, dene the \concave conjugate" of F, \ F, as \ F () = inf fx? F (x)g ; : x:x Note that if F is concave, then \ [\ F ] = F. Similarly, dene the \convex conjugate" of F, [ F, as [ F () = sup fx? F (x)g ; ; x:x and note that [ [[ F ] = F if F is convex. It is straightforward to verify that for two functions F and G we have [ [F G]() = [ F () + [ G() ; which is the analogue of the convolution property for Fourier transforms. It is also interesting to study the mapping from b to b out in Theorem A, part (c). It is easy to verify that the function x is an eigenfunction of this mapping, in some sense. Furthermore it is not dicult to show that? \ [b out ]()? \ [b]() + [ [S]() : 3 Service Curve for Window Flow Control We return to the system illustrated in Figure 1. We will assume that network elements N f and N b guarantee service curves S f and S b respectively (not necessarily strict). 5

6 3.1 Service Curve for Window Flow Control We focus on nding a service curve for the rst buer and server. Given any t, let s = maxfs : s t and B (s) = g. From (7), it follows that R (s) = C? I (s) for s (s ; t), and therefore from (8) we have Since R, we have s R ()d C(t? s )? b cross (t? s ) : s R ()d [C(t? s )? b cross (t? s )] + : Noting that B (s ) =, it follows that the subsystem consisting of the second buer and second server strictly guarantees the service curve S, where S (x) = [Cx? b cross (x)] + : (9) Note, however, that the subsystem consisting of the rst buer and rst server does not necessarily guarantee the service curve ^S1 dened as ^S 1 (x) = [Cx? b cross 1 (x)] + ; (1) since the window ow control protocol may inhibit such a service guarantee. The next theorem identies a service curve that is guaranteed by the subsystem consisting of the rst buer and rst server. Before stating the theorem, we introduce some convenient notation. Given a function G, and a positive integer n, let G (n) be the n-fold convolution of G with itself, i.e. G (n) (x) = G G G(x) : {z } n In the case where n =, we dene G () (x) = ^(x). Theorem 1 (Service Curve for Window Flow Control) Suppose that the subsystem consisting of the second buer and second server guarantees the service curve S, and that network elements N f and N b guarantee the service curves S f and S b, respectively. Dene S loop (x) = ^S1 S f S S b (x). Then the subsystem consisting of the rst buer and rst server strictly guarantees the service curve S 1, where and Z + is the set of non-negative integers. S 1 (x) = min mz +f ^S1 S (m) loop(x) + mkg ; (11) Remark: It is interesting to note that that the service curve S 1 given in Theorem 1 is the \impulse response" of the linear feedback system depicted in Figure, under the \min-plus" algebra. To prove Theorem 1, we will use the lemma below. 6

7 Σ ^ S 1 S f S K* S b Figure : Window Flow Control as a Linear Feedback System. Lemma 1 For any xed t, under the hypothesis of Theorem 1, there exists a nite sequence of intervals (u n+1 ; t n ); (t n ; u n ); ; (u 1 ; t ), with t = t, such that B 1 (u n+1 ) = and: i u i+1 R 1 ()d ^S1 (t i? u i+1 ) 8i = ; ::; n (1) t i+1 R 1 ()d S f S S b (u i+1? t i+1 ) + K 8i = ; ::; n? 1 : (13) Furthermore, the lengths of these intervals satisfy the constraints: t i? u i+1 ; i = ; ::; n (14) u i+1? t i+1 K=C ; i = ; ::; n? 1 : (15) Proof of Lemma 1: Fix t and set t = t. We dene u i+1 in terms of t i : u i+1 = maxfmaxfs : s t i ; B 1 (s) = g; maxfu : u t i ; T (u) = Kgg : (16) Note that T (s) < K and B 1 (s) > 8s (u i+1 ; t i ) and hence using (6) and (8) we have i i R 1 ()d = C(t i? u i+1 )? I 1 ()d u i+1 u i+1 C(t i? u i+1 )? b cross 1 (t i? u i+1 ) : Remembering that R t i u i+1 R 1 ()d, and using the denition in (1), this implies i u i+1 R 1 ()d ^S1 (t i? u i+1 ) ; which proves (1). If B 1 (u i+1 ) = then set n = i. Otherwise, if B 1 (u i+1 ) >, then note that T (u i+1 ) = K. In this case, we will dene t i+1 in terms of u i+1 as follows. By the convolution theorem (Theorem B), the system consisting of the series cascade of N f, the second buer and second server, and N b guarantees a service curve of S f S S b (x). Thus, there exists t i+1 u i+1 such that R b ()d? i+1 R 1 ()d S f S S b (u i+1? t i+1 ) : (17) 7

8 Using equations (5) and (17), we get K = T (u i+1 ) = R 1 ()d? R b ()d R 1 ()d? S f S S b (u i+1? t i+1 )? i+1 R 1 ()d : (18) This proves (13). By construction, we have t i? u i+1, which is (14). Since service curves are nonnegative and R 1 is bounded above by C, we get from (18) K t i+1 R 1 ()d C(u i+1? t i+1 ) ; (19) which implies (15). By construction, t i+1 u i+. Hence by adding the non-negative quantity C(t i+1? u i+ ) to the right side of (19), we get u i+1? u i+ K=C >. Thus, since the system is completely empty at time and t is nite, the recursion must end in a nite number of iterations, i.e. B 1 (u i+1 ) = for some nite i. Thus, the construction results in a nite sequence of intervals as claimed in the lemma. } Proof of Theorem 1 : Fix any t >. Invoking Lemma 1, we have u n+1 R 1 ()d = nx i= nx i= i u i+1 R 1 ()d + ^S 1 (t i? u i+1 ) + = ^S1 (t n? u n+1 ) + ^S1 (t n? u n+1 ) + n?1 X i= n?1 X i= n?1 X i= n?1 X i= t i+1 ^S1 S (n) loop (t? u n+1) + nk S 1 (t? u n+1 ) : R 1 ()d fs f S S b (u i+1? t i+1 ) + Kg f ^S1 (t i? u i+1 ) + S f S S b (u i+1? t i+1 ) + Kg fs loop (t i? t i+1 ) + Kg Noting that B 1 (u n+1 ) =, this completes the proof. } Suppose that the delay in network element N f is upper bounded by f, and that the delay in N b is upper bounded by b. This would happen if N f and N b represent propagation delay. In this case, it is easy to show that for d = f or d = b, network element N d guarantees the service curve S d, where S d (x) = ^(x? d ). We use this fact in the next corollary, which considers ane burstiness constraints on the cross-trac. Corollary 1 Suppose b cross 1 (x) = a + a x, b cross (x) = b + b x, and the delay through N f and N b are bounded by f and b, respectively. Dene = f + b + [ a =(C? a )] + [ b =(C? b )], and = maxf a ; b g. Then the system consisting of the rst buer and rst server strictly guarantees a service curve of S1 (x) where S 1 (x) = min mz + Qm 1 (x) ; () 8

9 and for m 1: Q 1(x) = (C? a )(x? a ) + = ^S1 (x) ; (1) C? a Q m 1 (x) = (C? )(x? m? a C? a ) + + mk : () The next corollary assesses the impact of the window size K. Dene ( (C? K )(f + b ) + a + C?a C? = b b, if a > b (C? )( f + b ) + C? b C? a a + b, otherwise. Corollary Suppose b cross 1 (x) = a + a x and b cross (x) = b + b x, and the delay through N f and N b is upper bounded by f and b respectively. Then the system consisting of the rst buer and rst server strictly guarantees a service curve of S1 (x), where if K K then ( Q S 1 (x) = 1(x) = ^S1 (x), if a > b (4) minfq 1(x); Q 1 1(x)g, otherwise, and if K < K then lim inf x!1 S 1 (x) x (3) = K : (5) Remark: Corollary implies that a window size equal to the sum of the burstiness of the cross-trac at the rst server ( a ), the burstiness of the cross-trac at the second server ( b ), and the user bandwidth delay product ((C? )( f + b )) is sucient to guarantee the maximum guaranteed throughput of C?. 3. Buer Requirements for Window Flow Control If the input stream R is characterized by a burstiness constraint, an upper bound on B 1 (t) can be derived from Theorem A and Theorem 1. An obvious upper bound for B (t) is the window size K, which can be achieved if the delay through network elements N f and N b is allowed to be zero. However, it is sometimes possible to derive an upper bound on B (t) which is smaller than the window size K, even under no assumptions on R. To illustrate this, we now consider the case where N f and N B represent constant delays of f and b, respectively. We will derive an upper bound on B (t) using two facts. First, note that in this case we have Thus, T (t? f ) =?f R 1 ()d??(f + b ) R ()d K : B (t) =?f fk + = K? R 1 ()d??(f + b ) t?( f + b ) R ()d R ()dg? R ()d R ()d : (6) 9

10 Second, in this case, the time elapsed from the time a piece of trac leaves the rst server until the acknowledgement for it returns to the rst server is at least f + b. Thus, for any s and t such that t? s f + b we have s R 1 ()d K : (7) Since R 1 is bounded above by the capacity of the server C, from (7) it follows that for any s and t such that t? s f + b we have Both inequalities (6) and (7) are due to [6]. s R 1 ()d minfc(t? s); Kg : (8) Theorem Suppose that N f and N b represent constant delays f and b, respectively, and that the cross-trac I at the second server satises the burstiness constraint (8). Then the amount of trac B (t) in the second buer satises where B (t) maxfb 1 ; B g ; (9) B 1 = max x:x f + b fminfcx; Kg? [Cx? b cross (x)] + g B = K? [C( f + b )? b cross ( f + b )] + : Proof Fix t and dene u = maxfs : s t; B (s) = g. If t? u > f + b then B (s) > for all s (t? f? b ; t) and hence R ()d t? f? b = C( f + b )? I ()d t? f? b Thus, in this case it follows from (6) that B (t) B. If t? u f + b then using (8) it follows that B (t) = u?f R f ()d? [C( f + b )? b cross ( f + b )] + : (3) R ()d u = R 1 ()d? [C? I ()]d u? f u minfc(t? u); Kg? [C(t? u)? b cross (t? u)] + B 1 : } As an example, suppose that b cross (x) = b + b x, and b =(C? b ) K=C f + b. In this case Theorem yields that B (t) b + b (K=C) : 1

11 R (t) B 1 (t) I 1 (t) R 1 (t) C R 1 (t) N 1 f B (t) I (t) R (t) C R (t) I n-1 (t) R n-1 (t) C R n-1 (t) f N n-1 B n (t) I n (t) R n (t) C R n (t) I 1 (t) N 1 b I (t) I n-1 (t) b N n-1 I n (t) Figure 3: Several Servers in Tandem. As claimed, the upper bound above may be considerably smaller than the window size K. Smaller upper bounds on B (t) can be obtained if R 1 is more strongly regulated. One way to achieve this is to assume a lower bound on the cross-trac at the rst server. In this case, if there is insucient cross-trac at the rst server to achieve this, the rst server could \pretend" there is additional cross-trac. Perhaps a better approach is to have the rst server directly control the burstiness of R 1. This is a topic for future research. 4 Several Servers in Tandem We now investigate the several server case as shown in Figure 3 where we have n cascaded buer-server pairs. The model is analogous to the two server case and precise denitions are omitted here for brevity. In order to analyze this multiserver system, we may apply Theorem 1 recursively as follows. Given a burstiness constraint for the cross-trac at the n th server, a service curve S n strictly guaranteed by the n th buer-server pair is implied, analogous to (9). Theorem 1 then implies that a service curve S n?1 is strictly guaranteed by the (n? 1) th buer-server pair. This process is repeated until a strict service curve guarantee of S j is derived for the j th buer-server pair for all j. The end-to-end service curve for the entire system is then (S 1 S S n ) (S f 1 S f Sn)(x) f by Theorem B. The following corollary follows by using this method. Corollary 3 Consider the tandem conguration of n buer-server pairs as illustrated in Figure 3, where network elements N f j and Nj b have maximum delays f j and j b, respectively. Suppose b cross i (x) = i + i x for all i = 1; ::; n, where = 1 = = ::: = n. If K j j + j+1 + (C? )( j b + j f ) 8j = 1; ::n? 1; then the system consisting of the j th buer and the j th server strictly guarantees a service curve of S j (x) = [? j + (C? )x] + 8 j = 1; :::; n : Furthermore, the entire system guarantees a service curve of S total (x) = (C? )(x? P n j? P n?1 j=1 C? j=1 j f ) +. If R is such that R t s R ()d + (t? s) for all s t, where C?, then the total end-to-end delay is bounded above by D total, where P n j= D total = n?1 j X C? + j f : (31) j=1 11

12 The results in Corollary 3 improve previously reported bounds, where we showed that P n?1 j=1 K j = O(n ) was sucient for a guaranteed throughput of C? [3] and for a maximum total end-to-end delay bounded by O(n ) [4]. References [1] R. Cruz, \A Calculus for Network Delay, Part I: Network Elements in Isolation," IEEE Transactions on Information Theory, Vol. 37, No. 1, Jan [] R. Cruz, \Quality of service guarantees in virtual circuit switched networks," IEEE Journal on Selected Areas in Communications, Vol. 13, No. 6, Aug [3] R. Cruz, \Guaranteed throughput for window ow control in a multi-tasking environment," Proceedings of the 1991 Conference on Information Sciences and Systems, John Hopkins University, March [4] R. Cruz, `Tandem queues with nite buers in a multi-tasking environment: performance guarantees via rate control," Proceedings of the 199 Conference on Information Sciences and Systems, Princeton University, March 199. [5] E. L. Hahne, \Round robin scheduling for fair ow control in data communications networks," IEEE JSAC, vol. 9, no. 7, Sept. 1991, pp [6] S. Khorsandi and A. Leon-Garcia, "Robust non-probabilistic bounds for delay and throughput in credit-based ow control," Proceedings IEEE INFOCOM '96, vol., pp [7] H. T. Kung, T. Blackwell, and A. Chapman, \Credit update protocol for ow controlled ATM networks: statistical multiplexing and adaptive credit allocation," Proceedings of SIGCOMM'94, pp [8] S. M. Meerkov and F. Top, \Analysis and synthesis of asymptotically reliable serial production lines," Technical Report CGR-4, Dept. of EECS, University of Michigan, Nov [9] A. K. Parekh and R. G. Gallager, \A generalized processor sharring approach to ow control in integrated services networks: the single node case," IEEE/ACM Trans. Networking," vol. 1, no. 3, June 1993, pp [1] H. Sariowan, \A service-curve approach to performance guarantees in integratedservice networks," Ph.D. thesis, Dept. of Electrical & Computer Engineering, UCSD, June