Shreehari Anand Bodas, Michel Mandjes, Liron Ravner
{"title":"具有非平稳到达和未观察到的停顿的服务系统的统计推断","authors":"Shreehari Anand Bodas, Michel Mandjes, Liron Ravner","doi":"arxiv-2311.16884","DOIUrl":null,"url":null,"abstract":"We study a multi-server queueing system with a periodic arrival rate and\ncustomers whose joining decision is based on their patience and a delay proxy.\nSpecifically, each customer has a patience level sampled from a common\ndistribution. Upon arrival, they receive an estimate of their delay before\njoining service and then join the system only if this delay is not more than\ntheir patience, otherwise they balk. The main objective is to estimate the\nparameters pertaining to the arrival rate and patience distribution. Here the\ncomplication factor is that this inference should be performed based on the\nobserved process only, i.e., balking customers remain unobserved. We set up a\nlikelihood function of the state dependent effective arrival process (i.e.,\ncorresponding to the customers who join), establish strong consistency of the\nMLE, and derive the asymptotic distribution of the estimation error. Due to the\nintrinsic non-stationarity of the Poisson arrival process, the proof techniques\nused in previous work become inapplicable. The novelty of the proving mechanism\nin this paper lies in the procedure of constructing i.i.d. objects from\ndependent samples by decomposing the sample path into i.i.d.\\ regeneration\ncycles. The feasibility of the MLE-approach is discussed via a sequence of\nnumerical experiments, for multiple choices of functions which provide delay\nestimates. In particular, it is observed that the arrival rate is best\nestimated at high service capacities, and the patience distribution is best\nestimated at lower service capacities.","PeriodicalId":501330,"journal":{"name":"arXiv - MATH - Statistics Theory","volume":"82 6","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Statistical inference for a service system with non-stationary arrivals and unobserved balking\",\"authors\":\"Shreehari Anand Bodas, Michel Mandjes, Liron Ravner\",\"doi\":\"arxiv-2311.16884\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study a multi-server queueing system with a periodic arrival rate and\\ncustomers whose joining decision is based on their patience and a delay proxy.\\nSpecifically, each customer has a patience level sampled from a common\\ndistribution. Upon arrival, they receive an estimate of their delay before\\njoining service and then join the system only if this delay is not more than\\ntheir patience, otherwise they balk. The main objective is to estimate the\\nparameters pertaining to the arrival rate and patience distribution. Here the\\ncomplication factor is that this inference should be performed based on the\\nobserved process only, i.e., balking customers remain unobserved. We set up a\\nlikelihood function of the state dependent effective arrival process (i.e.,\\ncorresponding to the customers who join), establish strong consistency of the\\nMLE, and derive the asymptotic distribution of the estimation error. Due to the\\nintrinsic non-stationarity of the Poisson arrival process, the proof techniques\\nused in previous work become inapplicable. The novelty of the proving mechanism\\nin this paper lies in the procedure of constructing i.i.d. objects from\\ndependent samples by decomposing the sample path into i.i.d.\\\\ regeneration\\ncycles. The feasibility of the MLE-approach is discussed via a sequence of\\nnumerical experiments, for multiple choices of functions which provide delay\\nestimates. In particular, it is observed that the arrival rate is best\\nestimated at high service capacities, and the patience distribution is best\\nestimated at lower service capacities.\",\"PeriodicalId\":501330,\"journal\":{\"name\":\"arXiv - MATH - Statistics Theory\",\"volume\":\"82 6\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-11-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Statistics Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2311.16884\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Statistics Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2311.16884","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Statistical inference for a service system with non-stationary arrivals and unobserved balking
We study a multi-server queueing system with a periodic arrival rate and
customers whose joining decision is based on their patience and a delay proxy.
Specifically, each customer has a patience level sampled from a common
distribution. Upon arrival, they receive an estimate of their delay before
joining service and then join the system only if this delay is not more than
their patience, otherwise they balk. The main objective is to estimate the
parameters pertaining to the arrival rate and patience distribution. Here the
complication factor is that this inference should be performed based on the
observed process only, i.e., balking customers remain unobserved. We set up a
likelihood function of the state dependent effective arrival process (i.e.,
corresponding to the customers who join), establish strong consistency of the
MLE, and derive the asymptotic distribution of the estimation error. Due to the
intrinsic non-stationarity of the Poisson arrival process, the proof techniques
used in previous work become inapplicable. The novelty of the proving mechanism
in this paper lies in the procedure of constructing i.i.d. objects from
dependent samples by decomposing the sample path into i.i.d.\ regeneration
cycles. The feasibility of the MLE-approach is discussed via a sequence of
numerical experiments, for multiple choices of functions which provide delay
estimates. In particular, it is observed that the arrival rate is best
estimated at high service capacities, and the patience distribution is best
estimated at lower service capacities.