ORDINARY CLT AND WLLN VERSIONS OF L=λW - Queueing limit theorems

Sep 14, 2026

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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.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

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Introduction and summary

Functional vs ordinary limit theorems

Framework and notation

Main results and theorem statements

Stationarity-based CLT and WLLN versions

# ORDINARY CLT AND WLLN VERSIONS OFL=λW*+ PETER W.GLYNN\#AND WARD WHITT The 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. 1.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. In 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] The 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 *Received September 11,1985;revised October 5,1987. AMS 1980 subject classification.Primary:90B22;Secondary:60K25.IAOR 1973 subject classification.Main:Queues. OR/MS Index 1978 subject classification.Primary:694 Queues/Limit theorems. Key words.Queueing theory,conservation laws,Litle's law,central limit theorems,law of large numbers,random sums,inverse processes. Supported 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 0364-765X/88/1304/0674S01.25 Copyright ◎1988,The Institute of Management Sciences/Operations Research Society of America ORDINARY CLT AND WLLN VERSIONS OF L=λW SLLN(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. As in [7],we use the standard L=λW fra

FAQs

What does this paper show about the queueing principle L=λW?

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.

Why is stationarity an extra condition in the main results?

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.

How do ordinary limit theorems relate to functional-limit-theorem counterparts?

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.

Sep 14, 2026

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