[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-268103-105":59,"doc-detail-268103-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","ordinary-clt-and-wlln-versions-of-lw-queueing-limit-theorems","ORDINARY CLT AND WLLN VERSIONS OF L=λW - Queueing limit theorems","","ORDINARY CLT AND WLLN VERSIONS OF L=λW develops central-limit-theorem and weak-law-of-large-numbers analogs of Little’s law by interpreting the queueing identity L=λW as a relationship among ordinary limit theorems. The results are established under strict stationarity, extending previous functional-central-limit-theorem work via different arguments. The paper derives CLT behavior for queue-length and related processes, analyzes conservation-style mappings, and studies how CLTs/SLLNs/WLLNs correspond—or fail to correspond—across ordinary and functional forms. It also presents equivalences between strong functional laws and provides conditions ensuring WLLN implies SLLN.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/ordinary-clt-and-wlln-versions-of-lw-queueing-limit-theorems/268103/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/ordinary-clt-and-wlln-versions-of-lw-queueing-limit-theorems/268103.png","ImageObject",300,407,{"name":92,"@type":93},"Kurz","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does this paper show about the queueing principle L=λW?","Question",{"text":113,"@type":114},"It provides ordinary central limit theorem (CLT) and weak law of large numbers (WLLN) versions of L=λW, interpreted as relationships among probabilistic limit theorems for queueing quantities.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Why is stationarity an extra condition in the main results?",{"text":118,"@type":114},"The paper shows that corresponding relations among the CLTs (and their functional counterparts) require strict stationarity; without it, earlier functional-limit results cannot be transferred in the same way.",{"name":120,"@type":111,"acceptedAnswer":121},"How do ordinary limit theorems relate to functional-limit-theorem counterparts?",{"text":122,"@type":114},"Some implications do not hold in general: a CLT need not imply a functional CLT, and a WLLN need not imply a functional WLLN. However, strong law functional versions are shown to be equivalent in the paper’s results.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},268103,1789428435,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":46,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":31,"preview_page_count":4},2336478945635,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","# ORDINARY CLT AND WLLN VERSIONS OFL=λW*+\n\nPETER W.GLYNN\\#AND WARD WHITT  \nThe familiar queueing principle expressed by the formula L=λW(Little's law)can beinterpreted as a relation among strong laws of large numbers(SLLNs).Here we provecentral-limit-theorem(CLT)and weak-law-of-large-numbers(WLLN)versions of L=λW.For example,if the sequence of ordered pairs of interarrival times and waiting times is strictlystationary and satisfies a joint CLT,then the queue-length process also obeys a CLT with arelated limiting distribution.In a previous paper we proved a functional-central-limit-theoremversion of L=λW,without stationarity,by very different arguments.The two papers highlightthe differences between establishing ordinary limit theorems and their functional-limit-theoremcounterparts.  \n1.Introduction and summary.In [7]we established a functional-central-limit-theorem(FCLT)version of the fundamental queueing formula L=λW [12],[14].Weshowed that the time-average of the queue-length process obeys a FCLT if thecustomer-average waiting time obeys a FCLT jointly with the customer-averageinterarrival time,and we described the limits.This was accomplished using thecontinuous mapping theorem and related arguments in the setting of weak convergenceon the function space D[0,∞],as in [1]and [16].Since FCLTs tend to hold in allthestandard situations in which ordinary central limit theorems(CLTs)hold,the FCLTsin [7]seem quite satisfactory for practical purposes(e.g.,applications to queueingparameter estimation;see [7]-[9D.Nevertheless,it is natural to ask if it is possible toestablish corresponding relations among the associated CLTs;here we show that it ispossible,provided that we add the extra condition of stationarity.  \nIn addition to the extending the queueing relation L=λW,the results here havegeneral probabilistic interest.In particular,we establish new asymptotic results forrandom sums and inverse processes.This paper complements [16],in which it is shownthat functional limit theorems are preserved under various mappings on the functionspaces D(0,∞)and D[0,∞]×D(0,∞)such as composition and inverse.Here similarresults are obtained for ordinary limit theorems.For example,for inverse processes,§3here is an analog of §7 of [16]  \nThe importance of this paper given [7]hinges on the relation between ordinary limittheorems(CLTs,SLLNs and WLLNs)and their functional-limit-theorem counterparts(FCLTs,FSLLNs and FWLLNs).Consequently,we also address this issue here(§2 and 6).We show that a CLT plus stationarity need not imply a SLLN or aFWLLN(Example 1 in §6).As a consequence,a WLLN need not imply a FWLLN,and a CLT need not imply a FCLT.On the other hand we show that SLLNs andFSLLNs are equivalent (Theorem 4).We also show that a FCLT need not imply a  \n*Received September 11,1985;revised October 5,1987.  \nAMS 1980 subject classification.Primary:90B22;Secondary:60K25.IAOR 1973 subject classification.Main:Queues.  \nOR/MS Index 1978 subject classification.Primary:694 Queues/Limit theorems.  \nKey words.Queueing theory,conservation laws,Litle's law,central limit theorems,law of large numbers,random sums,inverse processes.  \nSupported by the National Science Foundation under Grant No.ECS-8404809 and by the U.S.Army underContract No.DAAG29-80-C-0041 at the University of Wisconsin-Madison.Stanford University.AT&T Bell Laboratories  \n0364-765X/88/1304/0674S01.25  \nCopyright ◎1988,The Institute of Management Sciences/Operations Research Society of America  \nORDINARY CLT AND WLLN VERSIONS OF L=λW  \nSLLN(Example 2 in §6).As a consequence,we obtain the well-known result that aWLLN need not imply a SLLN.However,as an important tool for establishing ourqueueing results,we show that a WLLN does imply a SLLN under the extra conditionsof stationarity,nonnegativity and finite mean(Theorem 5).Obviously much of thisbackground has been discovered before,e.g.,Theorem 4,but the importance makes abrief explicit treatment worthwhile.  \nAs in [7],we use the standard L=λW fra","cbCait3xMJDsPs7c","https://ap.wps.com/l/cbCait3xMJDsPs7c","pdf",1046383,"English","# Introduction and summary\n## Functional vs ordinary limit theorems\n## Framework and notation\n# Main results and theorem statements\n## Stationarity-based CLT and WLLN versions","[{\"question\":\"What does this paper show about the queueing principle L=λW?\",\"answer\":\"It provides ordinary central limit theorem (CLT) and weak law of large numbers (WLLN) versions of L=λW, interpreted as relationships among probabilistic limit theorems for queueing quantities.\"},{\"question\":\"Why is stationarity an extra condition in the main results?\",\"answer\":\"The paper shows that corresponding relations among the CLTs (and their functional counterparts) require strict stationarity; without it, earlier functional-limit results cannot be transferred in the same way.\"},{\"question\":\"How do ordinary limit theorems relate to functional-limit-theorem counterparts?\",\"answer\":\"Some implications do not hold in general: a CLT need not imply a functional CLT, and a WLLN need not imply a functional WLLN. However, strong law functional versions are shown to be equivalent in the paper’s results.\"}]","ORDINARY CLT AND WLLN VERSIONS OF L=λW - Queueing limit theorems | PDF"]