{"title":"Energy-performance tradeoffs in server farms with batch services and setup times","authors":"Thu Le-Anh , Tuan Phung-Duc","doi":"10.1016/j.peva.2025.102468","DOIUrl":"10.1016/j.peva.2025.102468","url":null,"abstract":"<div><div>Data centers consume a large amount of energy, much of which is wasted due to idle servers. Turning off idle servers might be an effective power-saving solution; however, there is a trade-off between energy savings and system performance. Hence, we propose a setup queueing model with a batching policy that allows servers to process a set of jobs simultaneously to minimize power consumption while maintaining acceptable performance. We consider an M/M/<span><math><mrow><mi>c</mi><mo>/</mo></mrow></math></span>SET–BATCH queue, a multi-server batch service queue with a fixed batch size and setup times, and some variants, including systems in which idle servers delay before turning off or systems in which the batch size is dynamic. We analyze the steady-state probabilities and system performance of the M/M/<span><math><mrow><mi>c</mi><mo>/</mo></mrow></math></span>SET–BATCH system and its variants. Our analysis of the M/M/<span><math><mrow><mi>c</mi><mo>/</mo></mrow></math></span>SET–BATCH system with lower computational complexity is made possible by utilizing the special structure of the model. In addition, we use simulations to compare the M/M/<span><math><mrow><mi>c</mi><mo>/</mo></mrow></math></span>SET–BATCH model with some other variants with different setup time distributions. The results suggest that the model performs better when the setup time has a larger coefficient of variation. Our results indicate that the batching policy enhances the system performance, especially when we allow servers to be idle before turning them off.</div></div>","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"168 ","pages":"Article 102468"},"PeriodicalIF":1.0,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Coupled queues with server interruptions: Some solutions","authors":"Herwig Bruneel, Arnaud Devos","doi":"10.1016/j.peva.2024.102466","DOIUrl":"10.1016/j.peva.2024.102466","url":null,"abstract":"<div><div>We study three different <em>discrete-time</em> queueing systems, which accommodate two types of customers, named type 1 and type 2. New customers arrive independently from slot to slot, but the numbers of arrivals of both types in any slot are possibly mutually dependent; their joint probability generating function (<em>pgf</em>) is <span><math><mrow><mi>A</mi><mrow><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>. The service times of all customers are deterministically equal to one time slot.</div><div>We first consider a scenario (<em>Option</em> <span><math><mi>A</mi></math></span>) with <em>one single server</em> which is to be shared by the two customer types. Here, we assume that type-1 customers have <em>absolute service priority</em> over type-2 customers. Moreover, the server is subject to <em>random server interruptions</em>, which occur independently from slot to slot. We derive a functional equation for the steady-state joint pgf <span><math><mrow><mi>U</mi><mrow><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> of the numbers of type-1 and type-2 customers in the system. Relying on the application of Rouché’s theorem, we are able to explicitly solve the functional equation for <em>arbitrary</em> arrival pgfs <span><math><mrow><mi>A</mi><mrow><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, but more elegant results are obtained for some specific choices of <span><math><mrow><mi>A</mi><mrow><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>.</div><div>Next, we focus on two different scenarios (<em>Option</em> <span><math><mi>B</mi></math></span> and <em>Option</em> <span><math><mi>C</mi></math></span>) where both customer types have their <em>own dedicated server</em>. Here, there are no service priorities involved. In Option <span><math><mi>B</mi></math></span>, the two servers experience <em>simultaneous</em> interruptions, whereas in Option <span><math><mi>C</mi></math></span>, <em>only one</em> of the servers is subject to interruptions. Again, we derive functional equations for the pgf <span><math><mrow><mi>U</mi><mrow><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>. Although solving these equations for arbitrary arrival pgfs <span><math><mrow><mi>A</mi><mrow><mo>(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></ms","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"167 ","pages":"Article 102466"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143181845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Formal error bounds for the state space reduction of Markov chains","authors":"Fabian Michel, Markus Siegle","doi":"10.1016/j.peva.2024.102464","DOIUrl":"10.1016/j.peva.2024.102464","url":null,"abstract":"<div><div>We study the approximation of a Markov chain on a reduced state space, for both discrete- and continuous-time Markov chains. In this context, we extend the existing theory of formal error bounds for the approximated transient distributions. In the discrete-time setting, we bound the stepwise increment of the error, and in the continuous-time setting, we bound the rate at which the error grows. In addition, the same error bounds can also be applied to bound how far an approximated stationary distribution is from stationarity. As a special case, we consider aggregated (or lumped) Markov chains, where the state space reduction is achieved by partitioning the state space into macro states. Subsequently, we compare the error bounds with relevant concepts from the literature, such as exact and ordinary lumpability, as well as deflatability and aggregatability. These concepts provide stricter than necessary conditions for settings in which the aggregation error is zero. We also present possible algorithms for finding suitable aggregations for which the formal error bounds are low, and we analyze first experiments with these algorithms on a range of different models.</div></div>","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"167 ","pages":"Article 102464"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143181844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Editorial: Special issue on Performance Analysis and Evaluation of Systems for Artificial Intelligence","authors":"Anshul Gandhi , Bo Jiang , Shaolei Ren","doi":"10.1016/j.peva.2024.102465","DOIUrl":"10.1016/j.peva.2024.102465","url":null,"abstract":"","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"167 ","pages":"Article 102465"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143182335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Job assignment in machine learning inference systems with accuracy constraints","authors":"Tuhinangshu Choudhury , Gauri Joshi , Weina Wang","doi":"10.1016/j.peva.2024.102463","DOIUrl":"10.1016/j.peva.2024.102463","url":null,"abstract":"<div><div>Modern machine learning inference systems often host multiple models that can perform the same task with different levels of accuracy and latency. For example, a large model can be more accurate but slow, whereas a smaller and less accurate can be faster in serving inference queries. Amidst the rapid advancements in Large Language Models (LLMs), it is paramount for such systems to strike the best trade-off between latency and accuracy. In this paper, we consider the problem of designing job assignment policies for a multi-server queueing system where servers have heterogeneous rates and accuracies, and our goal is to minimize the expected inference latency while meeting an average accuracy target. Such queueing systems with constraints have been sparsely studied in prior literature to the best of our knowledge. We first identify a lower bound on the minimum achievable latency under any policy that achieves the target accuracy <span><math><msup><mrow><mi>a</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> using a linear programming (LP) formulation. Building on the LP solution, we introduce a Randomized-Join-the Idle Queue (R-JIQ) policy, which consistently meets the accuracy target and asymptotically (as system size increases) achieves the optimal latency <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mtext>LP-LB</mtext></mrow></msub><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span>. However, the R-JIQ policy relies on the knowledge of the arrival rate <span><math><mi>λ</mi></math></span> to solve the LP. To address this limitation, we propose the Prioritize Ordered Pairs (POP) policy that incorporates the concept of <em>ordered pairs</em> of servers into waterfilling to iteratively solve the LP. This allows the POP policy to function without relying on the arrival rate. Experiments suggest that POP performs robustly across different system sizes and load scenarios, achieving near-optimal performance.</div></div>","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"167 ","pages":"Article 102463"},"PeriodicalIF":1.0,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143181843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dimensioning leaky buckets in stochastic environments","authors":"Peter Buchholz , András Mészáros , Miklós Telek","doi":"10.1016/j.peva.2024.102461","DOIUrl":"10.1016/j.peva.2024.102461","url":null,"abstract":"<div><div>Leaky buckets are commonly used for access control in networks, where access control stands for traffic regulation at the ingress of the network. In network calculus, which is often applied for performance analysis or dimensioning of networks, leaky buckets are the model behind piecewise linear arrival curves that specify an input bound to a network. In this paper we present the analysis of leaky bucket based access control under stochastic arrivals using fluid queues, when the access control is implemented by possibly more than one leaky buckets. This results in methods to dimension parameters of access control for different stochastic arrival processes including correlated arrivals. The approach is one step to bridge the gap between classical stochastic analysis using queuing networks and deterministic analysis using network calculus. Results are presented for stochastic arrival processes using numerical methods and for measured arrivals using trace driven simulation.</div></div>","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"167 ","pages":"Article 102461"},"PeriodicalIF":1.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143181842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Enabling grant-free multiple access through Successive Interference Cancellation","authors":"Asmad Bin Abdul Razzaque, Andrea Baiocchi","doi":"10.1016/j.peva.2024.102460","DOIUrl":"10.1016/j.peva.2024.102460","url":null,"abstract":"<div><div>Internet of Things (IoT) is stirring a surge of interest in effective methods for sharing communication channels, with nodes transmitting sporadic, short messages. These messages are often related to control systems that collect sensor data to drive process actuation, such as in industries, autonomous vehicles, and environmental control. Traditional approaches that dominate wireless and cellular communications prove most effective when dealing with a limited number of concurrently active nodes, sending relatively large volumes of data. We address a different scenario where numerous nodes generate and transmit short messages according to non-periodic schedules. In such cases, random multiple access becomes the typical approach for sharing the communication channel. We propose a general modeling framework that enables the investigation of the impact of Successive Interference Cancellation (SIC) on two of the main random access paradigms, namely Slotted ALOHA (SA) and Carrier-Sense Multiple Access (CSMA). The key varying parameter is the target Signal to Interference plus Noise Ratio (SINR) at the receiver, directly tied to the spectral efficiency of the adopted coding and modulation scheme. Two different regimes are highlighted that bring the system to work at relative maxima of the sum-rate. We further investigate the impact of different transmission power settings and imperfect interference cancellation. Leveraging on the insight gained in the saturated node scenario, an adaptive algorithm is defined for the dynamic case, where the number of backlogged nodes varies over time. The numerical results provide evidence of a significant potential for grant-free multiple access, calling for practical algorithms to translate this promise into feasible realizations.</div></div>","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"167 ","pages":"Article 102460"},"PeriodicalIF":1.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143182242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Analysis of a queue-length-dependent vacation queue with bulk service, N-policy, set-up time and cost optimization","authors":"P. Karan, S. Pradhan","doi":"10.1016/j.peva.2024.102459","DOIUrl":"10.1016/j.peva.2024.102459","url":null,"abstract":"<div><div>Due to the extensive applications of bulk service vacation queues in manufacturing industries, inventory systems, wireless sensor networks for deducing energy consumption etc., in this article, we analyze the steady-state behavior of an infinite-buffer group arrival bulk service queue with vacation scenario, set-up time and <span><math><mi>N</mi></math></span>-threshold policy. Here the customers arrive according to the compound Poisson process and the server originates the service process with minimum ‘<span><math><mi>a</mi></math></span>’ customers and can give service to maximum ‘<span><math><mi>b</mi></math></span>’ customers at a time. We adopt batch-size-dependent service time as well as queue-length-dependent vacation duration which improve the system’s performance significantly. The <span><math><mi>N</mi></math></span>-threshold policy is proposed to awaken the server from a vacation/dormant state where the service station starts the set-up procedure after the accumulation of pre-decided ‘<span><math><mi>N</mi></math></span>’ customers. Using the supplementary variable technique, firstly, we derive the set of system equations in the steady-state. After that, we obtain the bivariate probability generating functions (pgfs) of queue content and size of the departing batch, the queue content and type of vacation taken by the server at vacation completion epoch and also the single pgf of queue content at the end of set-up time. We extract the joint distribution from those generating functions using the roots method and derive a simple algebraic relation between the probabilities at departure and arbitrary epoch. We also provide assorted numerical results to validate our proposed methodology and obtained theoretical results. The impact of the system parameters on the performance measures is presented through tables and graphs. Finally, a cost optimization function is provided for the benefit of system designers.</div></div>","PeriodicalId":19964,"journal":{"name":"Performance Evaluation","volume":"167 ","pages":"Article 102459"},"PeriodicalIF":1.0,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142723450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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