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Table 3: A comparison based on multi-server queuing system (M/M/c) λ µ L theoretical L measured 0.02 0.1 0.2 0.198 0.025 0.067 0.375 0.367 0.04 0.056 0.72 0.728 0.022 0.03 0.734 0.722 0.02 0.027 0.74 0.739 0.02 0.04 0.5 0.492 0.033 0.067 0.5 0.495 The Kolmogorov-Smirnov test computed D statistics of 0.16 and P statistic of 0.95 similar to the earlier the case of single-server queuing system indicating that the data set of L measured and L theoretical has no significant difference. Thus, the (M/M/c) queuing system developed for this case also conforms to the theoretical results. We conclude the theoretical validation of the DES model built to be used in this thesis with the above three validations. The results are not exactly equal to the theoretic results because of the slight numerical inaccuracies of the implementations of probabilistic distributions. In addtion, the multi-client class systems fall under multi-server multi-class queuing systems. Well known exact theoretical results are not available so far for such systems, so that we limited our validation to multi-server queuing systems. With this result we can justify that the implementation of the constructs of DES model, including scheduling of multiple resource units and queuing is valid. 5. Simulation settings Using the above generalized DES model, we setup a simulation system to apply and validate the proposed nonlinear control theoretic approaches in this thesis. 5.1. Workload profiles The workloads, a multi-client class system may face cannot be generalized. The workload a system can manage depends on the capacity of resources, management requirements and performance objectives. The workload profile for a system with CPU as the shared resource may differ from a system with concurrent threads as a shared resource. In addition, the workloads are time-varying, instead of staying constant for entire period of operations. This characteristic is not only limited to software systems, but to other physical systems as well. As a consequence, control engineering provides set of well-established input signals to validate the performance of the control systems. They are as follows: Assume, W n is the nominal workload that system receive. Impulse input signal: Formally, W impulse (k) = 1 when k = 0 and k 0. i.e, the impulse input signal increases the workload to some value greater than W n for a single sample period. In a real workload this can be considered as a workload spike for a very short time period. However, such spikes for very short periods of time may not affect the performance attributes (e.g., average response time) drastically, consequently the impulse input signal may not be useful for the validations of the control systems of software systems. Step input signal: Step input signal models a sudden jump in the workload from W n to some value W step and staying at that value for a more than a single sample period. This is one 8
of the widely used input signals to validate the performance of the control systems in control engineering. In addition, most of the applications of feedback control in software systems, including multi-client class systems have used step workload changes to validate the performance and resource management capabilities. This is because, such workload changes of even a single client class in a multi-client system for a long period of time affects the performance attributes (e.g., response time) under control. As a consequence, the control system is forced to redistribute the available recourses among client classes, in order to achieve the required performance objectives. The delay in response to such workload variations may cause large transient responses and temporal instabilities in the system. Therefore, this is a significantly difficult load variation to handle [2, 3, 4, 5]. Ramp input signal: Ramp input linearly increases the workload from W n to W ramp during sometime interval. This signal models a gradual increase of workload instead of instantaneous increment of workload compared to step input signal. The main advantage of these input signals is given a linear model of a system, there are wellknown design and analysis techniques available from control theory to compute performance specifications and behavior. Consequently, after constructing a linear model of a system we can investigate/prove the load variations that the system can maintain without leading to instabilities. However, a linear model of a system is an estimation of its behavior (not 100% accurate representation), so that these theoretical evaluations may not be correct 100%. Further, this is also true for systems demonstrating nonlinearities such as the system under investigation in this thesis. As a consequence, we have to mention that the combinations of workload input signals (in particular, step input profiles) in time varying fashion are used as heuristics to validate and compare the performance of the control systems. 5.2. Total resource amount and resource reservation time distribution The following settings will be used as an abstract representation of the multi-client class system in the simulations. The settings will remain the same unless otherwise specified. The total amount of resources simulated S total = 30. The processing time of each resource unit is selected from a uniform distribution as follows : 1 r(x) = for r min ≤ x ≥ r max r max − r min (5) = 0 for x < r min and x > r max (6) Where, r min = 100 ticks and r max = 700 ticks. The selection of the above settings is done, in order to achieve the tractability of resource allocations among client classes under different experiment conditions. The r min and r max , were selected after careful investigation of system outputs under different workload conditions. That is when the system is running close to the full capacity the system output should remain within some bounds, according to theoretical and practical system behavior. The Figure 2 shows a comparison when 30 resource units are allocated to two client classes with 30 req/sec workloads for each class. When the selected bounds r min = 100 and r max = 700 ticks, maintain the system in steady state under the applied resource settings. However, under the same settings, when the bounds are r min = 100 and r max = 900 ticks, the steady state behavior is highly variable/unstable. This is because the variability around the average response time leads to large transient response in the system. To avoid such behaviors the resource capacity and the workload intensity have to be selected depending on the bounds. For the workloads rates and the resources we selected to evaluate, r min = 100 ticks and r max = 700 are suitable bounds. 9
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