De Bock, Jasper

De Cooman, Gert

Lopatatzidis, Stavros

English

Ghent University Academic Bibliography

First steps towards Little’s Law with imprecise probabilitiesStavros Lopatatzidis, Jasper De Bock, Gert de CoomanSYSTeMS research group, Ghent University, Belgium{Stavros.Lopatatzidis,Jasper.DeBock,Gert.deCooman}@UGent.beLittle’s Law Little’s Law is described bythe equation L=λW , where L is the averagenumber of items in a queuing system, λ is theaverage number of arriving items per unit timeand W is the average waiting time of an item.Our queuing setting We examine the lawfor a discrete-time, single-server queue wherethe arrivals and the servicing (departures)happen according to imprecise Bernoulliprocesses: forward irrelevant arrivals occurat each discrete time point with probabilityinterval [a,a] and, similarly, forward irrelevantdepartures occur at each discrete time pointwith probability interval [d,d]. Arrivals anddepartures are assumed to be strongly inde-pendent. We make two additional assumptionsregarding the properties of the queue. Thefirst one is that upon arriving, an item needsto remain in the queue till served. Secondly,departure is characterised by the FIFO (first infirst out) principle.Our results Using the framework of coherentlower (and upper) previsions, our main result isa relation between the lower (and upper) pre-vision of the remaining waiting time of the lastitem in the queue (Wt) and the lower (and up-per) prevision of the number of items in thequeue (Xt) at any given time point t. Morespecifically, we prove that P(Xt) = dP(Wt)and P(Xt) = dP(Wt).AbstractWe consider a queue with maximum capacity N, where, at each discrete time point, departure is assumed to occur prior to arrival.We can treat this queuing system as an Imprecise Markov Chain. Due to forward irrelevance, its credal set is very intuitive. It consistsof all the precise models that can be constructed as depicted below. Arrivals occur with probability akt ∈ [a,a], which can differ betweenstates and time points. The same holds for departures (dkt ∈ [d,d]). In these expressions, kt corresponds to the number of items attime point t. Finally, p1 is the initial probability distribution, taking values in some credal setM1.NN · · ·· · · NNN−1NNN−1· · ·· · ·· · ·· · ·kk+ 1· · ·· · ·· · ·k· · ·· · · kkk−1k+ 1k· · ·k−1· · ·· · ·· · ·· · ·01· · ·· · ·· · ·0 · · ·· · · 01010extended state space(conditioning on events)local transition modelinitial statep 1(0)p1(k)p1 (N)(1−a01)a01dk1(1−ak1)dk1ak1+(1−dk1)(1−ak1)(1−dk1 )ak1dN1(1−aN1)(1−dN1)+ dN1aN1d0na0nd0n(1−a0n)(1−d0n)a0n(1−d0n)(1−a0n)dknakndkn(1−akn)(1−dkn)akn(1−dkn)(1−akn)dNnaNndNn(1−aNn)(1−dNn)aNn(1−dNn)(1−aNn)The queue we considert t+ 1 t+ 2xxxx−1x−1x−1x−2Dt+1 =0D t+1=1Dt+2 = 0Dt+2=1Dt+2 = 0Dt+2=1Due to the FIFO principle and the assumption that an item stays in thequeue till served, the waiting time (Wt) of the last item of the queue, at anytime point t, will depend on the number of items that exist in the queue atthat time (Xt) and on the sequence of departures from t+ 1→ ∞.Due to forward irrelevance, the lower prevision of Wt can be obtained bymeans of the following expression, which closely resembles the law ofiterated expectation, as used in precise-probabilistic approaches.P(Wt) = P(∑x∈XI{x}(Xt)P(Wt|Xt = x)). (1)In this expression, Ix is the so-called indicator of x.The next step consists in realising that P(Wt|Xt = x) depends on the depar-ture process only. It is the time until x departures have occurred. We willuse P(W (x)) as an alternative, shorthand notation for P(Wt|Xt = x). Then,P(W (x)), similarly to (1), can be written as followsP(IdC(Dt+1)P(W (x)|Dt+1 = 0)+ Id(Dt+1)P(W (x)|Dt+1 = 1)). (2)We also know that the remaining waiting time W (x) of the item under study,at time t, is actually the remaining waiting time at time t+ 1 increased byone. This means that the two components of (2) becomeP(W (x)|Dt+1 = 0) = P(Wt|Xt = x,Dt+1 = 0) = P(Wt+1|Xt+1 = x)+ 1 (3a)P(W (x)|Dt+1= 1) = P(Wt|Xt = x,Dt+1= 1) = P(Wt+1|Xt+1= x−1)+1 (3b)Taking into account that the probability tree is infinite with respect to itslength (t→ ∞), the item under study at t+ 1 waits until x departures occur ifDt+1 = 0, otherwise (Dt+1 = 1) it waits until x−1 departures occur. Hence(3a) and (3b) becomeP(W (x)|Dt+1= 0) = P(W (x))+1 , P(W (x)|Dt+1= 1) = P(W (x−1))+1 (4)where P(W (0)) = 0.We understand also that the following inequality holdsP(W (x))≥ P(W (x−1)). (5)Now (2) can be reformulated as followsP(W (x)) = P(Id(Dt+1)P(W (x)|Dt+1 = 1)+ IdC(Dt)P(W (x)|Dt+1 = 0)) == P(Id(Dt+1)P(W (x−1))+ IdC(Dt+1)P(W (x)))+ 1== dP(W (x−1))+ (1−d)P(W (x))+ 1(6)where the second equality is derived from (4) and the third from (5).Therefore, (6) results inP(W (x)) = dP(W (x−1))+ (1−d)P(W (x))+ 1⇒ P(W (x)) = xd. (7)By plugging (7) into (1) we have thatP(Wt) = P(∑x∈XI{x}(Xt)P(Wt|Xt = x))= P(∑x∈XI{x}(Xt)xd)= P(∑x∈XI{x}(Xt)x)1d=P(Xt)d⇒ P(Xt) = dP(Wt).Similarly we can prove that P(Xt) = dP(Wt).Heuristic proof of the main resultDifferent types of independence More stringent indepen-dence assumptions can be imposed on the arrival and depar-ture processes, such as epistemic independence or strongindependence. With sensitivity analysis, we still get the sameresult as for forward irrelevance. P(Xt)d is minimized for a andd, meaning that the expected remaining waiting time is min-imized when we take the lowest arrival probability and thehighest respective one for departure. This leads to the samebounds under either epistemic independence or strong inde-pendence. Also, strong independence between arrivals anddepartures can be weakened to epistemic independence.Experimental work We also provide some experimentalresults regarding the lower and upper prevision of the num-ber of items in the queue (P(Xt)), the number of items inthe queue conditional on an entrance (P(Xt|entr)) and theremaining waiting time of the last item conditional on an en-trance (P(Wt|entr)), as t→ ∞. The P(Wt|entr) is our approachto studying the imprecise average waiting time. We useda queue of maximum capacity 5, departures occur prior toarrivals, where the probability interval of arrival is [0.6,0.8]and the respective one of departure is [0.5,0.7]. Additionally,we chose the vacuous model to be our initial model.0 10 20 30 40 50 60 70 80 90 100 110012345678910t’Prevision(P)’ P(Xt)P(Xt|entr)P(Wt|entr)P(Xt)P(Xt|entr)P(Wt|entr)Additional results

First steps towards Little's Law with imprecise probabilities

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First steps towards Little's Law with imprecise probabilities

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