# Analysis of End-to-End Response Times of Multi-Tier Internet Services

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4 i =,, N Each service tier i is modeled by a processing sharing queue and draws its service times from a general probability distribution with an average service time of β s,i time units for i =,, N The mean response time of the service is modeled as the sojourn time of the request in the system Let S (k) i be the sojourn time of the k-th visit to level i, and M = T T N Then, the expected sojourn time S of an arbitrary request arriving to the system is given by M+1 X = S k=1 S (k) + T NX X i i=1 j=1 S (j) i Note that the system is modeled such that it satisfies the conditions of a product-form network First, the arrivals occur according to a Poisson process with a rate that is not state dependent This is not unrealistic in practice, since arrivals from a large number of independent sources do satisfy the properties of a Poisson process [14] Second, the basic services are modeled by mixtures of processor sharing and infinite server queues with a general service distribution which does not depend on the number of requests at that node Finally, since the sequence in which the service nodes are visited is fixed, and thus does not depend on the state of the system, the network is of product-form 4 THE MEAN RESPONSE TIME AND ITS VARIANCE In the previous section we have seen that the queueing network is of product-form Consequently, when L c,i and L s,i denote the stationary number of requests at caching tier i and service tier i for i = 1,, N, respectively, we have È(L c,i = l c,i, L s,i = l s,i; i =,, N) = NY NY c,i = l c,i) s,i = l s,i), i=è(l i=è(l with l c,i and l s,i =, 1, for i =,, N From this expression, the expected sojourn time at the entry node and the service nodes can be determined First, define the load on the entry nodes by ρ c, = λp β c, for the caching tier and ρ s, = (M + 1)(1 p )λβ s, for the service tier Similarly, the load for the basic services is given by ρ c,i = (1 p )p iλβ c,i and ρ s,i = (1 p )(1 p i)λβ s,i for i = 1,, N Then, by Little s Law, the expected sojourn time S c,i at caching tier i is given by S c,i = β c,i, whereas the expected sojourn time S s,i at service tier i is given by S s,i = β s,i/(1 ρ s,i) for i =,, N Combining all the expressions for the expected sojourn time at each tier in the network, we derive that the expected response time is given M+1 X by S = k=1 S (k) + T NX X i i=1 j=1 S (j) i = p S c, + NX (1 p ) S s, + i pi S c,i + (1 p i) S s,i i=1 T (M + 1)βs, = p β c, + (1 p ) + 1 ρ s, (1 p ) NX i=1 T i β s,i p iβ c,i + (1 p i) 1 ρ s,i Let us now focus our attention to the variance of the response times It is notoriously hard to obtain exact results for the variance Therefore, we approximate the total sojourn time of a request at service tier by the sum of M + 1 independent identically distributed sojourn times Moreover, we approximate the variance by imposing the assumption that the sojourn times at the entry node and the sojourn times at the service nodes are uncorrelated In that case, we have Îar S =Îar M+1 X k=1 M+1 X =Îar k=1 S (k) + T NX X i i=1 j=1 S (k) +Îar S (j) i X N T X i i=1 j=1 S (j) i To approximate the variance of the sojourn times at tier, we use the linear interpolation of Van den Berg and Boxma in [29] to obtain the second moment of the sojourn time of an M/G/1/PS node We adapt the expression by considering the M +1 visits together as one visit with a service time that is a convolution of M + 1 service times Let c s,i denote the coefficient of variation of the service times at service tier i for i =,, N Then, we have that the variance of the sojourn time S s, at service tier is approximately given byîar S s,(m + 1), with Îar S s,i(k) = (K + 1)c 2 s,i ρs,i βs,i 2 ρ s,i 1 ρ s,i 2 (K + 1)βs,i + `(K + 1) 2 (K + 1)c 2 s,i 1 ρ s,i 2β 2 s,i (1 ρ s,i) 2 2β 2 s,i ρ 2 s,i (1 ρs,i)(eρ s,i 1 ρ s,i) In principle, the same expression for the variance of service tiers i can be used with M =, yielding the expressionîar S s,i(1) for i = 1,, N Recall that a single request to tier can lead to multiple interactions between tier and tier i This introduces additional variability W i due to various reasons, such as context switching, request and response processing We estimate this variability generated by the random variable T i by Wald s equation [3] as follows Îar W i =Îar T i jîar S s,( T i) +Îar S s,i(1) + (1 p ) 2 ( T i + 1)β s, + T i p iβ c,i + (1 p i) 1 ρ s, β s,i 1 ρ s,i «2 ff Let β (2) c,i denote the second moment of the service times at caching tier i for i =,, N Then, the varianceîar S c,i for caching tier i is given by Îar Sc,i = β (2) c,i β2 c,i Finally, by combining all the expressions for the variances of the sojourn times at each node in the network, we derive that the variance of the response time is given by Îar S p 2 Îar S c, + (1 p ) 2 jîar S s,(m + 1) + NX i=1 p 2iÎar S c,i + (1 p i) 2Îar S s,i(1) +Îar W i ff Note that the expressions exhibit a lot of structure and clearly show how each tier contributes to the mean and the variance This makes the expressions highly flexible so that changes in the resource configuration of the application can be easily accommodated for For instance, suppose that service tier i is replaced with

5 β s, c 2 s, β s,1 β s,2 Îar s S Îar S Îar % Table 1: Response time variances of a queueing network with general service times at the entry nodes and two asymmetrically loaded single-server service nodes β s, c 2 s, β s,1 β s,2 Îar s S Îar S Îar % Table 2: Response time variances of a queueing network with general service times at the entry node and two asymmetrically loaded multi-server service nodes an FCFS queueing system with c i servers having exponentially distributed service times with mean β s,i (note that this retains the product form solution) Then, only S s,i andîar S s,i have to be changed To this end, let ρ s,i = (1 p )p iλβ s,i/c i, and define the probability of delay π i by π i = (ciρs,i)c c i i 1 X (c iρ s,i) l (1 ρ s,i) + (ciρs,i)c i 1 c i! l! c l= i! Then, the mean sojourn time S s,i at service tier i is given by S s,i = β s,i (1 ρ s,i)c i π i + β s,i, and the variance of the sojourn timeîar S s,i at service tier i is approximated πi(2 πi)β2 s,i byîar Ss,i c 2 s,i (1 + β2 s,i ρs,i)2 5 VALIDATION WITH SIMULATIONS In this section we assess the quality of the expressions of the mean response time and the variance that were derived in the previous section We perform some numerical experiments to test validity of the expressions against a simulated system In this case, the mean response time does not need to be validated, because the results are exact due to [25] Therefore, we can restrict our attention to validating the variance only We have performed extensive numerical experiments to check the accuracy of the variance approximation for many parameter combinations This was achieved by varying the arrival rate, the service time distributions, the asymmetry in the loads of the nodes, and the number of servers at the service nodes We calculated the relative error by Îar % = 1% `Îar S Îar s S /Îar s S, whereîar s S is the variance based on the simulations We have considered many test cases We started with a queueing network with exponential service times at the entry node and two service nodes at the back-end In the cases where the service nodes were equally loaded and asymmetrically loaded, we observed that the relative error was smaller than 3% and 6%, respectively We also validated our approximation for a network with five service nodes The results demonstrate that the approximation is still accurate even for very highly loaded systems Based on these results, we expect that the approximation will be accurate for an arbitrary number of service nodes The reason is that cross-correlations between different nodes in the network disappear as the number of nodes increases Since the cross-correlation terms have not been included in the approximation (because of our initial assumptions), we expect the approximation to have good performance in those cases as well Since the approximation for different configurations of the service nodes turned out to be good, we turned our attention to multiserver service nodes We carried out the previous experiments with symmetric and asymmetric loads on the multi-server service nodes while keeping the service times at the entry nodes exponential Both cases yielded relative errors smaller than 6% Finally, we changed the service distribution at the entry node Table 1 shows the results for a variety of parameters, where the coefficient of variation for the service times at the entry nodes is varied between (deterministic), 4 and 16 (Gamma distribution) These results are extended in Table 2 with multi-server service nodes If we look at the results, we see that the approximation is accurate in all cases To conclude, the approximation covers a wide range of different configurations and is therefore reliable enough to obtain the variance of the response time 6 VALIDATION WITH EXPERIMENTS In the previous section, we demonstrated the accuracy of our mean and variance expressions using simulations In this section, we validate our model with two well-known benchmarks: RUB- BoS, a popular bulletin board Web application benchmark, and TPC-App, the latest benchmark from the Transactions Processing Council modeling service oriented multi-tiered systems The choice of these two applications was motivated by their differences in their behavior We validate our model by running these benchmarks on a Linuxbased server cluster for the following resource configurations: (i) when the application is deployed with a single application server and a database server, (ii) when the application is deployed with a front-end cache server, an application server, and a database server, and (iii) when the application is deployed with a single application server, a database cache server, and a database server We first present our experimental setup followed by the validation results 61 Experimental Setup We hosted the business logic of the applications in the Apache Tomcat/Axis platform and used MySQL 323 for the back-end database servers We ran our experiments on Pentium III machines with 9 Mhz CPU and 2 GB memory running a Linux 24 kernel These servers belonged to the same cluster and network latency between the clusters was less than a millisecond In our experiments, we varied the arrival rate and measured the mean end-to-end response and its variance and compared the measured latency with

10 A similar approximate analysis can be performed when a buffer of size N is added in front of the application When K customers are already in the system, the requests in excess of K are buffered and wait their turn to enter the system in an FCFS manner However, when the buffer is full, requests are again rejected The mean response time S ac,n, when admission control is applied with a buffer size of N, is then approximated by S ac,n È(L () = )q K l=1 N X l= qk l+1 l + q K 1 X K È(L (1) l = ) λ q l + K λ qk NX j=1 qk j, q K 1 where the expressionè(l (a) = ) is defined as X K N a X È(L (a) qk j 1 = ) = q l + q K q K 1 l= j=1 We can study the influence of the buffer size using the results for admission control systems with and without a buffer given in the previous paragraphs Figure 8(a) shows the influence of the buffer size N as iso-loss curves, ie, all combinations of N and K for which the blocking probabilityè(l () = N + K) is a constant p The results are for a 3-tier system with one service tier at the frontend, and two multi-server FCFS service tiers at the back-end The parameters used for this model are λ = 1, β s, = 3, β s,1 = 16, and β s,2 = 9 The results show that only three customers are allowed in the system to obtain a blocking probability of 5 when the buffer size is less than two When the buffer size is larger than two, only two customers are allowed Consequently, to guarantee that the blocking probability is less than 5, the threshold K must be greater than three when the buffer size N is less than two, and otherwise, K must be greater than two Since differences in the buffer size and threshold limit influence the mean response time, Figure 8(b) shows the delay curves corresponding to the values of the iso-loss curves as a function of N Thus, for instance, Figure 8(b) shows that for a blocking probability of 5, the mean response time increases linearly when the buffer size N increases The graphs of Figure 8 show the importance of selecting an appropriate buffer size with its corresponding blocking probability When the blocking probability is high, the number of allowed customers in the application will be small, so that queueing mainly occurs in the buffer in front of the application When the blocking probability is small, the buffer in front of the application is hardly used, so that the number of admitted requests is high with as result that queueing occurs mainly within the application 73 SLA negotiation In this section, we demonstrate how to compute performance measures that can be used in effective Service Level Agreement (SLA) negotiation SLA negotiation is a common problem that arises when an application is composed of not just its own business logic and databases but also using services provided by external companies For example, many small Internet retailers use amazoncom s Web service storage service 7 and order fulfillment service 8 to build their own application Now, let us see how our model can be used to perform these SLA negotiations Consider a 3-tier system with one service tier at the front-end, called the application server (AS), and two external β 5 s, E[S] = β s,1 (a) Negotiation space under the constraint [S] < 5 (b) Negotiation space under the constraint SD[S] < 2 Figure 9: The SLA negotiation space for all β s,1 and β s,2 service tiers that are part of different domains Assume that the AS has an SLA with its end-users stating that the mean duration for obtaining a response is less than 5 seconds with a maximum standard deviation of 2 seconds Moreover, suppose that requests of users arrive according to a Poisson process at the AS with rate λ = 1 The AS has a mean service time of β s, = 1 seconds, and the external service tiers are single-server FCFS nodes with exponentially distributed service times The key question for effective SLA negotiation is: What combination of SLAs with the other domains leads to the desired response times? More specifically, one could ask: What is the SLA negotiation space of the AS? Mean response times: The total response time can be split up in the sojourn time S s, at the AS, and the sojourn times S s,1 and S s,2 at the external service tiers The results of Section 4 yield that the mean sojourn time at the AS equals S s, = 43 seconds Since the mean response time is constrained by 5 seconds, a request can spend on average up to a maximum of 457 seconds at the other services The other services handle just one request at a time, ie, if there are more requests at those service tiers, requests have to wait and will be served according to an FCFS discipline The results in Section 4 enable us to compute all combinations of the mean service times β s,1 and β s,2 for which the total response time is at most 457 seconds On first glance, one would expect that the region of these service

11 β s, E[S] = 32 E[S] = 16 E[S] = 8 E[S] = 4 E[S] = 2 E[S] = β s, β s, (a) Iso-curves for the mean [S] SD[S] = 8 SD[S] = 4 SD[S] = 2 SD[S] = β s,1 (b) Iso-curves for the standard deviation SD[S] Figure 1: SLA iso-curves for [S] and SD[S] times is bounded by a linear curve, since higher service times at one tier must result in lower service times at the other Figure 9(a) shows that this is not the case, and that the boundary is non-linear This negotiation space, together with the costs of providing a specific service time, can be utilized by the AS to minimize the total costs by negotiating an SLA with both service tiers using the optimal β s,1 and β s,2 Standard deviation of the response times: By applying the same procedure as in the previous paragraphs, we can determine the negotiation space of the AS with the restriction that the standard deviation of the total sojourn time is at most 2 seconds By using the results of Section 4, we obtain the varianceîar S s, of the sojourn time at the AS Subsequently, we can compute all combinations of β s,1 and β s,2 such that their standard deviation, when added to Îar S s,, is at most 2 seconds Figure 9(b) shows the region of these service times To adhere to the two restrictions simultaneously, ie, the expectation of the sojourn time is at most 5 seconds and the standard deviation is at most 2 seconds, we intersect both regions to keep the smallest region with combinations of β s,1 and β s,2 In order to study the effects of the constraints on the parameter regions, we have constructed iso-curves for the upper bounds on the mean S and the standard deviation SD[S] Figure 1(a) and (b) show these iso-curves that depict all combinations of β s,1 and β s,2 such that S and SD[S], respectively, have a specific constant value The figure shows that a more strict constraint on S results in a more linear iso-curve In such a case, the service times are that low so that the sojourn times almost equal the service times, and therefore almost no queueing occurs The case for a less strict constraint results in an iso-curve that almost has a square shape This is explained by the fact that the longer requests have to wait in one queue of the system, the shorter they have to wait in the other queue as long as the service time of the second queue is less than the service time of the first queue 8 CONCLUSIONS In this paper, we presented an analytical model for multi-tiered Internet applications Based on this model, we can estimate the mean response time of an application and also provide accurate approximations to its variance We verified the effectiveness of our approximations using numerical simulations resulting in a margin of error of less 6% Subsequently, we performed extensive experimentations with multiple industry-standard Web application benchmarks for a wide-range of resource configurations In our experiments we observed that the error margin of the model is usually less than 1% This demonstrates the accuracy of our model to capture the real behaviour of a wide range of Internet applications To the best of our knowledge, this is the first work that aims to model the variability in response times for multi-tiered Internet applications Our proposed model is highly flexible as it allows to easily accommodate for changes in the resource configuration Moreover, the model can also handle applications deployed with multiple caching tiers with high accuracy We also demonstrated how our model can be applied to SLA-driven resource provisioning, effective admission control, and SLA negotiation We believe that our model is highly beneficial since it deals with many realistic design problems that are faced by the designers and administrators of modern Internet systems 9 REFERENCES [1] Vogels, W: Learning from the Amazon technology platform ACM Queue 4(4) (26) [2] Shneiderman, B: Response time and display rate in human performance with computers ACM Computing Surveys 16(3) (1984) [3] Menasce, DA: Web server software architectures IEEE Internet Computing 7(6) (23) 7881 [4] Doyle, R, Chase, J, Asad, O, Jin, W, Vahdat, A: Web server software architectures In: Proceedings of USENIX Symposium on Internet Technologies and Systems (23) [5] Kamra, A, Misra, V, Nahum, E: Yaksha: A controller for managing the performance of 3-tiered websites In: Proceedings of the 12th IWQoS (24) [6] Abdelzaher, TF, Shin, KG, Bhatti, N: Performance guarantees for web server end-systems: A control-theoretical approach IEEE Trans Parallel Distrib Syst 13(1) (22) 896 [7] Chen, J, Soundararajan, G, Amza, C: Autonomic provisioning of backend databases in dynamic content Web servers In: Proceedings of the 3rd IEEE International Conference on Autonomic Computing (ICAC 26) (26) [8] Urgaonkar, B, Pacifici, G, Shenoy, P, Spreitzer, M, Tantawi, A: An analytical model for multi-tier Internet services and its applications In: Proceedings of the 25 ACM SIGMETRICS international conference on

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