Environment (E) IBP IBP IBP 2 N 2 N. server. System (S) Adapter (A) ACV

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1 The Adaptive Cross Validation Method - applied to polling schemes Anders Svensson and Johan M Karlsson Department of Communication Systems Lund Institute of Technology P. O. Box 118, Lund, Sweden Abstract The Adaptive Cross Validation (ACV) method is a general method for system performance optimization. It continuously investigates the performance of a number of simulation models in parallel to nd the currently best model. The best model and a number of alternative models are investigated next. This method can be used in the design phase to determine how a system should be designed, or it could be used during the operational phase to dynamically determine how the system should be controlled. A novel formalism, suitable for presentation of mathematical models in technical publications, is used to formally describe the studied system. The system is slotted, and consists of a shared server and a number of queues. A classical polling scheme (limited or gated) is used by the server at regular intervals to decide in which order the queues are served. The ACV method changes the polling frequency to optimize the system throughput even for bursty arrivals, in this case, one Interrupted Bernoulli Process for each queue. Both the steady state and the transient performance have been investigated. The advantage of using the ACV method over ordinary polling is demonstrated. 1 Introduction The issue of optimizing systems have been, and will be, one of the main objectives for many research studies. Usually the problem is the parameter setting in some algorithm, or any other mechanical or recursive computational procedure. Especially when the performance of the system is very dependent of the parameter setting and the optimal setting changes due to non stationary behavior of conditions that are out of control for the systems. In such cases a dynamic parameter setting, that follows the irregularities, would be a diserable property. Such a new dynamic method, called the Adaptive Cross Validation (ACV) method, is introduced and analyzed below. It mainly focus on the issue to provide better control of the transient behavior. To better understand the functionality and advantage of this method, it is, as an example, also applied to a polling system and some performance measures are presented. Polling systems could be used with dierent algorithms to decide how many jobs that should be attended from each queue in each cycle. These algorithms are often referred to as limited, gated and exhaustive search [1]. Several variants of the above basic polling schemes include probabilistic polling, and ic polling. For an explanation of these the reader is referred to [2]. In the studied example, the ACV method is applied to improve the performance of the limited scheme,

2 also be applied to many dierent systems within the telecommunication eld as well as outside. 2 Application description This section describes the usage of the ACV method for dynamic optimization of a resource sharing algorithm based on polling. Figure 1 shows a model of the application, which consists of three parts: an environment, a system, and an adapter. The environment consists of N sources, which are modeled by IBP processes, that generate jobs to the system. There are N queues in the system, one for each source, and a server, which is a shared resource. Each queue has space for L jobs. A polling algorithm is executed by the server ically with to determine the order in which the jobs in the queues are to be served. The ACV method is implemented by the adapter, which optimizes the server throughput by dynamically adjusting the. Environment (E) 1 IBP IBP IBP 2 N L 1 2 N System (S) Adapter (A) server ACV Figure 1: Model of the resource sharing application: environment (N job sources), system (N queues and one shared server) and adapter (ACV method). The application is dened in the mathematical model formalism MM [3], which has been designed to be compact, exact, and easy to read and understand, thereby making it easier to repeat experiments reported in technical publications. It uses a notation in between descriptions in programming languages, which are exact but require a lot of space, and abstract high level descriptions, which are short but leave the details to the reader. It provides people working with mathematical models with a \standardized" language. The overall structure of an application is described by a block scheme, which consists of a number of blocks. Each block, which is dened in a table, consists of variables and activities, which are triggered when certain conditions are fullled. Inter-block communication is handled by external variables, which are variables used by more than one block. A block is shown as a rectangle in a block scheme, while an external variable is illustrated by a symbol located close to the line that interconnects the blocks that use the variable. Figure 2 shows the block scheme for the application, that will be used and described below.

3 i Environment E t, System S n,h i im,t, τ, Adapter A τ,m Figure 2: Block scheme for the resource sharing application: environment E (N job sources), system S (N queues and one shared server) and adapter A (ACV method). 2.1 Environment block Table 1 denes the environment E. The arrival process from each source is an Interrupted Bernoulli Process (IBP), that is, for a geometrically distributed (on state) arrivals occur according to a Bernoulli process. This is followed by another (o state) which is also geometrically distributed, during which no arrivals occur. An IBP is governed by a two state Markov chain, where the time s between state changes are multiples of the slot time, S (the variable dened in the system block S). The state of source i, x i, can be either o or on, i = 1; :::; N. Given that a source is in the on state, the source will change to the o state with probability p OF F, or it will remain in the on state with probability 1? p OF F. If the source is in the o state, it will change to the on state with probability p ON, or it will remain in the o state with probability 1? p ON. A job is generated with probability when a source is in the on state. (x) x i = off; on [off] internal state, source i () 0:0 < N S [0:9] oered trac to the system [Erlang] C 2 = 1; 10; 100 [10] squared coecient of variation, arrivals (a) i = : i S ; 0:0 p OF F ; p ON ; 1:0 source #; probabilities; source trac 0:0 < t A [2] time when job arrivals may occur (y) y i = T; F (S; ) job arrival, source i (true,false) w (t S = t A ) : job arrivals r (1 i N S ) : f = =N S c (x i = on) : fy i = T ; p OF F = 1? (C 2? 1 + 3? 2 2 )=(1? + C 2 ) c (random < p OF F ) : fx i = offgg p ON = 1? (1? 2 + (1? p OF F ))=(1? ) (x i = off) : fy i = F ; t A = t S + S c (random < p ON ) : fx i = onggg Table 1: Environment block E: N sources generate jobs according to N independent Interrupted Bernoulli Processes with parameters and C 2. The total trac is called. Let be the interarrival time between jobs from a source, measured in number of slots. Equation 1 shows the expressions for the mean, E[], and the squared coecient of variation, C 2, of, derived in [4]. The burstiness in the job arrival process from one source is dened by C 2. The trac from one source, or the probability that a job arrives from one source,, is equal to the mean

4 over the mean length of the o and on s. Equation 2 shows the expressions for the = 1 case used to derive the expressions for p OF F and p ON in Table 1. E[] = 1 = p OF F + p ON p ON ; C 2 = V ar[] E[] 2 = 1 + [ p OF F (2? p OF F? p ON ) (p OF F + p ON ) 2? 1] (1) = p ON ; C 2 = p OF F (2? p OF F? p ON ) (2) p OF F + p ON (p OF F + p ON ) 2 Job arrivals may occur when the current time, t S, is equal to the arrival time t A. The system S is oered the trac = N. The p ON and p OF F probabilities are calculated each time a new slot begins to enable studies with time-varying trac. A random function, which returns a random number ( ), determines whether each individual source change state or not. The y i variable, which can be read by the S block, is set to T (true) when a job arrival from source i occurs. Note that job arrivals occur 2 after a slot starts, because the default, or initial, value for t A is equal to System block Table 2 denes the system S, which consists of N FIFO queues, one for each source, and a shared server. Each queue can hold L jobs. The system state, n i, denes the number of jobs in queue i, including the one currently being served, i = 1; :::; N. The server activities are divided into s of length slots. An activity list x j determines the server activity in slot j, j = 1; :::;. Possible activities in slot j are: executing the polling algorithm (x j =?1), idle (x j = 0) and processing a job from queue i (x j = i). A job's service time is equal to, the slot time. The server executes the polling algorithm at the beginning of a, at time t P, to determine the server activities during that. It takes O overhead slots to perform this task, which means that at most? O jobs from the queues can be processed during a. Table 2 shows that the rst O positions in x j are lled with minus one, and that k is set to the next slot to be scheduled. The longest queue is called ^i, and the number of jobs it contains is denoted by a^i. How many jobs that are selected from the longest queue, n, depends on a^i, the polling limit parameter, and the remaining slots to be scheduled,? k + 1. The parameter denes the maximum number of jobs that can be selected from a queue in each iteration of the algorithm. Two classical polling rules can be implemented: limited (1 < L) or gated ( = L). The remaining jobs to be scheduled in queue ^i, a^i, are updated before the next iteration starts. The procedure above is repeated until all slots have been scheduled (k > ), or until no more jobs remain to be scheduled (a^i = 0). Slots not used are lled with zeros to indicate server idleness. The current slot position within the is denoted by l. A job arrival from source i occurs when yi E is set to true. The job is accepted if there is less than L jobs in queue i. The h im variable, which is needed by the adapter A, records the arrival pattern to the queues in slot m. Job departures take place 2 before a slot ends, at time t D. A job departure from queue i = x l results in the updating of n i and s J, the number of jobs processed. The system performance is dened by the throughput, which is the probability that the server is busy processing jobs from the queues. Note that the maximum value is NL + O? 1 and not

5 when a slot starts, and job arrivals occur 2 after a slot starts. This implies that the queues can at most contain NL? 1 jobs when the polling algorithm is executed. It is clear that the performance of the polling algorithm depends on the and the polling limit. When the oered trac to the system is high, it is better to use longer s, because the relative overhead for executing the algorithm, O=, is then smaller. On the other side, when the oered trac is low, it is better to use shorter s, because a larger fraction of the slots are then used for job processing. The value that minimizes the variance between the queue lengths among the sources is one. This is only valid if the overhead O and the switch over time (time for the server to switch between two queues), do not depend on. It was decided that the should be optimized by the adapter A, primary for the case when = 1. (c) 1 N [5]; 1 L [10]; 0 O [1] # of queues; queue size; overhead slots 0:0 < [1:0] (A; E; ) slot time (x) 0 n i L [0] (A; ) # of jobs, queue i?1 x j N; h im = T; F (A; ); l = : j activity list; history logger; current slot () 1 + O NL + O? 1 [1 + O] (A; A) polling [slots] 1 L [1] maximum jobs selected/queue/iteration (a) 1 i;^i N; 1 j; k NL + O? 1 queue #; slot # within 0:0 t D [? 2]; 0:0 t P [0:0] departure time; polling time 1 n L; 1 m (; A); : a i = ni jobs/iteration; mega slot # (t) 0:0 t [0:0] (A; E; ) current time (p) 0:0 1:0; 0 s J ; s [0] throughput; # of processed jobs, slots w (t = t P ) : polling algorithm w (t = t D ) : job departure r (1 j O) : x j =?1; k = O + 1 c (1 x l N) : a i = n i ; ^i = maxargi a i fi = x l ; n i = n i? 1; s J = s J + 1g r (a^i > 0) ^ (k ) : f s = s + 1; = s J=s n = min(a^i ; ;? k + 1) l = l + 1; m = m + 1; t D = t + r (k j k + n? 1) : x j = ^i k = k + n; a^i = a^i? n; ^i = maxargi a i g r (k j ) : x j = 0; l = 1; t P = t + w (yi E = T ) _ (yi E = F ) : job arrivals c (yi E = T ) : n i = max(n i + 1; L); h im = yi E Table 2: System block S: N FIFO queues and a shared server. 2.3 Adapter block The ACV method [5], is a general method for system performance optimization. It can be used in the design phase to determine how a system should be designed, or it could be used during the operational phase to dynamically determine how the system should be controlled to optimize its performance. The method continuously investigates the performance of simulation models running in parallel with the real system under identical environmental conditions, as the real system, to nd

6 In this application, simulation models of the system S are executed in parallel, with the system, to optimize the server throughput. These simulation models are driven by exactly the same job arrival streams that have aected the system. It is the h im variable that provides the simulations with the job arrival streams. The simulation models are identical except for the polling algorithm used. This means that, by comparing the performance of the simulation models, a better than the one currently used by the system can be found. It takes several s to perform one iteration of this optimization procedure. Table 3 denes the adapter A, which consists of three activities performed during a mega. At time t 1, which occurs after the polling algorithm has executed, the number of jobs in the system queues are stored in a i ; the current slot in the mega, m S, is initialized; and the execution times are set. The start time for the next mega is stored in t 1. The length of a mega is + = 1 s of length 1 slots, where is the time it takes to perform the second activity, the simulations. At time t 2, which occurs when s has passed since time t 1, the performance of s, in this case = 3, the current 1, one shorter 2, and one longer 3, are investigated. The length of the alternative s depend on. The sim procedure takes the n, the simulation length 1, the initial system state a i, and the arrival pattern stored in h im, to compute the throughput n. The best procedure returns the best, ^. It approximates the ( 1 ; 1 ), ( 2 ; 2 ) and ( 3 ; 3 ) coordinates by a straight line, using the least squares method, if no single best was found. The longer, 3, is selected if the line has a positive slope; otherwise, the shorter is preferred if the slope is negative or if the line is horizontal. This means that the shorter is selected if 1 = 2 = 3. At time t 3, i.e. when slots has passed after time t 2, the selected can be used by the system S. (c) 3 [3]; 0 [10] # of simulation models; decision time [slots] : (a) 0:0 t 1 []; 0:0 t 2 ; t 3 ; a i = n S i execution times; # of jobs, queue i n ; ^ = : S ; 0:0 n 1:0 ; throughput 1 n ; m = : m S simulation #; mega slot # () 1 < [4]; 0:0 < [0:3] factor; alternative factor w (t S = t 1 ) : part one w (t S = t 2 ) : part two a i = n S i ; ms = 1; 1 = S 1 = S ; 2 = max( 1? 1 ; min( S )) t 1 = t S + ( + = 1 ) 1 S 3 = max( ; max( S )) t 2 = t S + 1 S r (1 n 3) : sim( n ; 1 ; a i ; h S im ; n) t 3 = t 2 +? 2 best( n ; n ; ^) w (t S = t 3 ) : part three S = ^ Table 3: Adapter block A: implements the ACV method.

7 The ACV method applied to the resource sharing application described in Section 2 was evaluated using simulation technique. To be able to get accurate results the simulations were executed for an extensive amount of times, and presented in the graphs below. The application model was initialized with the default values, specied in the tables in Section 2. In the simulations some of these parameters were altered, one or two at the time while the rest were kept at their default values. In this way it is possible to see the inuence of the individual parameters on the performance of the polling system. In general the new adaptive approach gives very good results compared to dierent static approaches. In the following these results are discussed and analyzed rho=n throughput rho= rho= rho=0.8 rho=0.9 probability Figure 3: The throughput versus for dierent loads (C 2 = 10), and the probability density function for the ACV method operated under the same conditions. 3.1 Steady state performance Figure 3 shows the throughput as a function of the, for three dierent trac loads. The three curves represent the system throughput for a relatively low load ( = 0:8), at presumed working load ( = 0:9) and at overload ( = N, the number of sources). All the results shown here have C 2 equals 10, the default value. As could be seen, the throughput for the system rises for longer s initially, due to the fact that there are more jobs waiting in the queues at the beginning of each than the length of the. As the s get longer there are not enough jobs in the queues to ll up a whole, so the system is hence forced to make plans for a whole without jobs to ll the last service slots within that. For the overload case, the throughput raises until it reaches the maximum. this point the throughput could be calculated as?o = NL?O NL = 0:98, shown in Figure 3. The lower part of Figure 3 shows the probability density function (pdf) of the s used while the At

8 C^2=1 throughput C^2=10 C^2= probability C^2= C^2=10 C^2= Figure 4: The throughput versus for dierent burstiness ( = 0:9), and the probability density function for the ACV method operated under the same conditions. ACV method was running under the same trac loads. The high probability of low length usage coincide with the highest throughput for each trac case respectively. This is, of course, the manner expected for a dynamic method aimed to optimize the throughput. The shape of the pdf is somewhat irregular, due to the fact that the only could be changed by steps,. For the overload case, i.e. job arrival in every slot, the pdf curve is not shown, however it has all the density in the equals NL. This is, however, an easier optimization problem since there is no stochastic behavior in the arrivals of jobs, i.e. the throughput is a monotone function of the. The burstiness of the arrival trac is also a factor that has an impact of the performance. This is shown in Figure 4, where the throughput is shown for three dierent settings of C 2, i.e. 1, 10 and 100. All three curves have the same load, in this case = 0:9, however the behavior is similar for other loads. As could be seen, the throughput decreases with increased burstiness, which is to be expected. The decrease is however rather substantial and the throughput is almost halved, for longer s, each time C 2 is increased a factor 10. This is due to the fact that for burstier trac patterns, there is a higher probability that the queues are empty when the polling algorithm is executed and hence a whole is scheduled without any jobs. This has more grave consequences for longer s. The lower part of Figure 4, which as above, shows the pdf for the has a slight tilt to lower s. This depends on the best procedure in Table 3, which gives priority to short s. The throughput for the ACV method is shown in Figures 3-4 as straight lines for each parameter combination. As could be seen, independent of the initial, the performance for the ACV method is almost identical to the throughput of the optimal. The ACV method has its advantages for trac patterns with smaller C 2, for the parameter settings shown. The performance

9 , which is discussed below. 3.2 Transient performance An important and essential issue is how the method reacts to sudden changes in the trac conditions. This is discussed and shown in the following. First the impact of a wrong chosen initial parameter setting for the is shown. As an extreme, the max and min values have been chosen as initial starting points. As shown in Figure 5 the method rather swiftly adjusts to a value close to the optimum (compare the curve for = 0:9 (C 2 =10) in Figure 3) time Figure 5: The versus time for two dierent initial settings of, where the adaptiveness of the ACV method is shown. In the upper curve the initial was set to its maximal value (NL + O? 1) and for the lower curve to its minimum value (1 + O). The impact of the value is clearly shown in this graph. Obviously, the initial NL is too large, hence it should at the next decision be decreased to NL? NL 1 which is shown in the gure. To further emphasize the dynamic behavior of the method, two cases where the trac changes its characteristics by steps are introduced. In the rst case, shown in Figure 6, the trac changes its mean arrival rate from its normal conditions ( = 0:9) to an overload condition at time instant 5000 and then back to normal conditions again at time instant This is to demonstrate how a sudden increase in trac intensity, restricted in time, is handled by the method. As could be seen, the throughput for the ACV method dramatically increases when the arrival intensity is increased compared to the optimal value (shown by the dashed line). When the trac alters back to normal conditions it takes some time for the method to change the to a new optimum value, as shown in Figure 6. During this time there are not a sucient number 1 The, in this case, only alternative investigated is? = NL? NL = 35.

10 throughput ACV static time x ACV static time x 10 4 Figure 6: The throughput for the ACV method and the optimal static case for a sudden change in (0:8! N! 0:8), and the used by these. (C 2 = 10) of jobs to ll the long s, and hence the throughput decreases substantially. However, after a short while the method has once again adapted to the new circumstances and the throughput is back to its origin. This, rather fast adjustment in the could be looked upon in the lower part of Figure 6. The second case consists of a change in the characteristic of the trac pattern, i.e. the load ( = 0:9) is unchanged while C 2 is altered from C 2 = 1 to C 2 = 100 at time instant 5000 and then back to normal conditions again at time instant As shown in Figure 7 the throughput performance for the ACV method (solid line) is as good as identical with the optimal throughput (dashed line) for the initial trac conditions, however after the change in the burstiness (C 2 ) the ACV method has an improved performance compared to the reference case. As for the change in the load () the ACV method very quickly adopts to the new conditions and gives a substantial improvement in total throughput. The rapid adjustment could be seen in the lower part of Figure 7, where the actual used by the system is shown. The edges, at the changes in the trac characteristics, are very distinct, which shows that the method is very dynamic without losing consistency. 3.3 General remarks So far nothing have been mentioned about the eect that the polling limit parameter has on the results. All results shown are evaluated using the limited strategy with = 1. Investigations have shown that the two strategies, limited ( = 1) and gated ( = L), have almost identical throughput characteristics. If they dier, it is only by a small fraction in favour of the limited strategy. The inuence of the value is o less importance for high C 2 values. The number of simulation models investigated in parallel,, was set to three during the experiments reported on. This number

11 throughput ACV 0.5 static time x static ACV time x 10 4 Figure 7: The throughput for the ACV method and the optimal static case for a sudden change in C 2 (1! 100! 1), and the used by these. ( = 0:9) could if desired be enlarged, which have the potential to increase the performance, especially the transient performance. However, this potentially enhanced performance have to be paid for by heavier calculations and thereby usually higher O values. Investigations, which are not shown in this paper due to space limitation, have indicated that there exist a trade o between small and large values. The outline from these studies is that a low value should be used for varying trac conditions, to optimize the transient behavior. While a large value would give a good steady state performance and hence be used in the design case. 4 Conclusions A new adaptive method to optimize dynamic systems, ACV, have been introduced. It has been applied to a resource sharing algorithm based on polling. The ACV method could however be applied to many dierent systems within the telecommunication eld as well as outside. The method is applied to a polling system, which is described by a new mathematical model formalism (MM). The formalism is designed to be compact, exact and easy to read and understand, making it easier to repeat experiments reported in technical publications. Extensive performance evaluations have been carried out and reported. The steady state performance of the ACV method gives a throughput close to that of the optimal static for the polling algorithm. The method was shown to rapidly adapt to changes in the environment, in this case, the mean load and burstiness of the arrival process. This results in an improved overall throughput performance. The main advantage of the ACV method is that no assumptions have to be made about the eect that the environment has upon the system. For queueing systems this would mean the arrival and service processes. In this way, a lot of time could be saved since

12 References [1] J. M. Karlsson, H.G. Perros and I. Viniotis,\Adaptive Polling Schemes for an ATM Bus with Bursty Arrivals', Computer Networks and ISDN Systems, Vol. 24, No. 1, [2] O. J. Boxma, H. Levy and J. A. Westrate, \Optimization of Polling Systems", in: King, Mitrani, Pooley, PERFORMANCE'90 (North Holland, Amsterdam, 1990), [3] A. Svensson, "The Mathematical Model Formalism", in preparation. [4] J. M. Karlsson, "Exact and Approximate Analytical Solutions of an ATM Buer Queueing Model - a Comparison", 2nd International Conference on Networks, paper E1.1, Singapore, [5] A. Svensson, "The Adaptive Cross Validation Method -Design and Control of Communication Systems", submitted for publication.