Static vs accumulating priorities in healthcare queues under heavy loads
SStatic vs accumulating prioritiesin healthcare queues under heavy loads
Binyamin Oz ∗ Seva Shneer † Ilze Ziedins ‡ Abstract
Amid unprecedented times caused by COVID-19, healthcare systems all over the worldare strained to the limits of, or even beyond, capacity. A similar event is experienced by somehealthcare systems regularly, due to for instance seasonal spikes in the number of patients.We model this as a queueing system in heavy traffic (where the arrival rate is approachingthe service rate from below) or in overload (where the arrival rate exceeds the service rate).In both cases we assume that customers (patients) may have different priorities and weconsider two popular service disciplines: static priorities and accumulating priorities. It hasbeen shown that the latter allows for patients of all classes to be seen in a timely manneras long as the system is stable. We demonstrate however that if accumulating prioritiesare used in the heavy traffic or overload regime, then all patients, including those with thehighest priority, will experience very long waiting times. If on the other hand static prioritiesare applied, then one can ensure that the highest- priority patients will be seen in a timelymanner even in overloaded systems.
We are currently seeing the effect COVID-19 has on healthcare services in vast majorityof countries in the world. Healthcare services are also under increasing pressure as demo-graphics change and populations age. Public health services are struggling, and sometimesfailing, to maintain services under this increasing load. Given this context of high demandfor tightly constrained resources it is instructive to reassess the rationales for prioritizationregimes currently in use, and contrast with other possibilities. Specifically, in this paper wecharacterize the performance of the very commonly used static priority regime, and contrastit with the recently proposed accumulating priority regime, under critical loadings.Healthcare systems have traditionally used static priority queues in a range of settingsfrom triage in an emergency departments (EDs), to organizing access to elective surgeries,such as hip and knee replacements [5],[1]. In a static priority regime, patients are assignedto a priority class, and must wait to be treated until all patients in higher priority classeshave been treated. Patients may sometimes be moved to a higher priority class if theircondition deteriorates, but in practice there is often no automatic mechanism for makingsuch transitions, and any adjustments may rely on patients proactively approaching theirhealthcare provider. Accumulating priority queues (APQs) have recently been proposed toovercome some of the inherent drawbacks of static priority queues in healthcare [13]. Inaccumulating priority queues, patients accumulate priority with time spent in the queue,at a rate that depends on their priority class, with higher priority patients accumulatingpriority faster than lower priority patients. Priority can accumulate linearly, or in a nonlinearfashion. Observational studies of behaviour in emergency departments have revealed thatin practice physicians may operate a regime that is similar to an APQ, with the likelihoodof being seen increasing more rapidly as waiting times approach threshold targets, see e.g.[5]. ∗ Hebrew University of Jerusalem † Heriot-Watt University ‡ The University of Auckland a r X i v : . [ c s . PF ] A p r he accumulating priority regime was first proposed by Kleinrock [10], who obtainedexpressions for the expected waiting times for all classes. A large-deviations principle hasbeen established in [15]. More recently, Stanford et al. [13] derived expressions for theLaplace Stieltjes Transform of the waiting time distribution, which can then be invertednumerically. A later paper [11], showed that a wide range of possible accumulation functions(including, for instance, exponential and log) have an equivalent linear regime, in the sensethat the order in which patients are seen is the same in both the nonlinear and linearformulation.This paper considers the performance of a single server queue with total arrival rate ρ and service rate 1, where ρ , the load on the server, either satisfies ρ ↑ ρ >
1. Theheavy traffic regime has been intensively studied, although not for the accumulating priorityqueue. When ρ > ρ > ρ is close to 1, then over relatively short periods of time it may be difficult to determinewhether ρ < ρ >
1. Thus there is a strong practical need to address the question ofhow best to organize patient prioritization in this transient regime.Indeed, we will see below that the two cases: a) ρ << ρ ↑ ρ > ρ <<
1, then theaccumulating priority regime ensures that all patients are seen in a timely manner, whileensuring health targets are met (if those targets are feasible). On the other hand, if ρ ↑ ρ >
1, then it is impossible to limit waiting times for all patients, and the static priorityregime provides a mechanism for ensuring that healthcare is still available to the most acute,and vulnerable, patients (see Fig. 2 below), whereas under APQ the expected waiting timesfor all classes increase (Fig. 1 below). For this scenario we propose below a mixture of thetwo prioritization schemes.Section 2 gives a detailed description of the models we consider. Section 3 considers thecase where ρ = 1 − (cid:15) , as (cid:15) ↓
0, while Section 4 considers the case where ρ >
1. In Section 5we discuss issue related to customers making strategic decisions. We conclude with a shortdiscussion of related models and potential future research directions in Section 6.
We consider a service system with a single server and N + 1 classes of customers. Customers(patients) of class i arrive according to a Poisson process with rate λ i and their priority classis associated with a positive real number b i for 1 ≤ i ≤ N + 1. The higher the number b i ,the higher the priority class, and without loss of generality we assume b > b > . . . > b N >b N +1 . Thus arrivals of class 1 are in the highest priority class and arrivals of class N + 1are in the lowest priority class.If a customer finds the server idle upon its arrival, the server immediately starts serving his customer. Priority is non-preemptive, that is, if the server is busy when a new customerarrives, the customer joins the waiting room regardless of her priority class. The room isassumed to be of infinite size. Whenever the server finishes serving a customer and thewaiting room is non-empty, the server starts to serve the customer with the highest currentpriority among those currently in the waiting room.We consider two different priority policies: • Static priorities (SP) : in this case a customer of class i has priority level b i whichdoes not change; • Accumulating priorities (AP) : in this case a customer of class i that spent s , s ≥ b i s .We assume, without loss of generality, that service times for customers are independentand exponentially distributed with mean 1. Therefore ρ = λ + . . . + λ N +1 (1)is the load on the system – the average amount of new work arriving per unit of time – and ρ < ρ = 1 − ε and ε ↓
0. The system is stable for any ε >
0, but, as ε decreasesto 0, the average number of customers waiting in the queue in the stationary regime (and,by Little’s formula, the average waiting time of a typical customer arriving in the system)tends to infinity. In the second scenario ρ > In this section we consider a sequence of systems indexed by ε > λ ( ε ) , . . . , λ N +1 ( ε ) are non-decreasingfunctions of ε > λ i ( ε ) ↑ λ i as ε ↓ i = 1 , . . . , N + 1, N +1 (cid:88) i =1 λ i ( ε ) = 1 − ε for all ε >
0, and N +1 (cid:88) i =1 λ i = 1 . An interesting special case of the setting above is the scenario where λ , . . . , λ N are fixedand λ N +1 ( ε ) = 1 − ( λ + . . . + λ N ) − ε , but we do not restrict ourselves to this case.Regardless of the service discipline chosen, the systems are stable for any ε >
0. Denoteby ( Q ε , . . . , Q εN +1 ) the vector of steady-state queue lengths and by ( W ε , . . . , W εN +1 ) thevector of steady-state waiting times (inclusive of the service time), for a particular value of ε . We also let Q εi = E ( Q εi ) and W εi = E ( W εi ) be the expected queue length and waitingtimes respectively, for all i . Regardless of the service discipline, Q ε + . . . + Q εN +1 = 1 − εε ∼ ε (2)as ε →
0. The total queue lengths thus increase to infinity, and we are interested in howqueue lengths, and waiting times, of the individual classes behave.
Consider first the SP priorities. Let σ i ( ε ) = (cid:80) ik =1 λ k ( ε ) , ≤ i ≤ N + 1, with σ ( ε ) = 0 . Let also σ i = lim ε ↓ σ i ( ε ). From Cobham [4], we obtain that the expected waiting times forthe priority classes are given by W εi = 1 − ε (1 − σ i − ( ε ))(1 − σ i ( ε )) + 1 , ≤ i ≤ N + 1 . (3) nd we can write W εN +1 = ε − ε + λ N +1 ( ε ) + 1 . Thus, we see that as ε → W εi → − σ i − )(1 − σ i ) + 1 < ∞ , ≤ i ≤ N. W εN +1 → ∞ and Q εi < ∞ , ≤ i ≤ N Q εN +1 ∼ /ε. Thus, as ε →
0, in the (SP) case, the expected queue lengths and waiting times for classes 1to N are bounded from above, while for class N +1 they grow without bound. The durationsof busy periods also increase without bound. For the AP case we can conclude from the Kleinrock formula ([10], see also [13]) that waitingtimes for all customers grow without bound as ε →
0, so that if the b i are held constant,this regime does not offer the same protection for the higher priority classes as SP does.Indeed we can prove the following exact statement. Lemma 1.
Consider an accumulating priority queue with N + 1 classes, and accumulationrates b > b > . . . > b N +1 . Then lim ε ↓ ε W εi = 1 /b iN +1 (cid:80) k =1 λ k /b k , ≤ i ≤ N + 1 . We present a proof of Lemma 1 below but first comment on its implications. The resultmay be interpreted as follows: a customer from class i entering service after waiting for time W i has, at that time, priority b i W i . Lemma 1 essentially states that the (AP) disciplinemakes all these priorities just before service equal on average, across classes, in heavy traffic.This is similar to the behaviour in heavy traffic of the MaxWeight protocol (see [14]) whichequates scaled queue lengths (we discuss this connection further below).Lemma 1 also implies that W εi → ∞ as ε ↓ i , and hencelim ε ↓ ε Q εi = lim ε ↓ ελ i ( ε ) W εi = λ i /b iN +1 (cid:80) k =1 λ k /b k , (4)which also means that lim ε ↓ Q εi (cid:80) N +1 k =1 Q εk = λ i /b iN +1 (cid:80) k =1 λ k /b k . (5) Proof of Lemma 1.
In order to simplify notation, in this proof we will drop thedependence of λ i and W i on ε . The formulas from [10] adapted to our setting are asfollows: W i = 1 /ε − − (cid:80) N +1 k = i +1 λ k (cid:16) − b k b i (cid:17) W k − (cid:80) ik =1 λ k (cid:16) − b i b k (cid:17) . We use a proof by induction on i = N + 1 , . . . ,
1. First write W N +1 = 1 /ε − − (cid:80) N +1 k =1 λ k (cid:16) − b N +1 b k (cid:17) = 1 /ε − − (cid:80) N +1 k =1 λ k + b N +1 (cid:80) N +1 k =1 λ k /b k = 1 /ε − ε + b N +1 (cid:80) N +1 i =1 λ k /b k , Class1 Class2 Class3
Figure 1: Numbers of customers of different classes against time in the system with accumulatingpriorities, with ε = 0 . which implies the statement of the lemma for i = N + 1. Assume now the statement is validfor all i ≥ j + 1 and let us prove if for i = j :lim ε ↓ ε W j = 11 − (cid:80) jk =1 λ k (cid:16) − b j b k (cid:17) − N +1 (cid:88) k = j +1 λ k (cid:18) − b k b j (cid:19) lim ε ↓ ε W k = 11 − (cid:80) jk =1 λ k + b j (cid:80) jk =1 λ k /b k − (cid:80) N +1 k = j +1 λ k (cid:16) − b k b j (cid:17) /b k (cid:80) N +1 k =1 λ k /b k = 1 (cid:80) N +1 k =1 λ k /b k − (cid:80) jk =1 λ k + b j (cid:80) jk =1 λ k /b k N +1 (cid:88) k =1 λ k /b k − N +1 (cid:88) k = j +1 λ k /b k + 1 b j N +1 (cid:88) k = j +1 λ k = 1 (cid:80) N +1 k =1 λ k /b k (cid:80) jk =1 λ k /b k + b j (cid:80) N +1 k = j +1 λ k (cid:80) N +1 k = j +1 λ k + b j (cid:80) jk =1 λ k /b k = 1 /b j (cid:80) N +1 k =1 λ k /b k . These results suggest that in the AP regime, the accumulation rates need to be adjustedif the system is subjected to increasingly heavy loads. A natural solution we propose is totake b i ( ε ) = c i /ε for fixed c i , { , . . . , N } , with b N +1 = c N +1 . This effectively applies the SP regime to the lowest priority class, while maintaining the positive benefits of the AP regimefor the remaining classes. More generally, we could consider regimes where b i ( ε ) = c i /ε forfixed c i , { , . . . , M } for any M ≤ N . If M < N , then the split between classes following AP and those following SP will still provide benefits to the higher priority classes. Whateverthe split, only the lowest priority class will experience delays that grow without bound. We illustrate the conclusions above with typical sample paths of a system with N = 2 (thereare therefore 3 classes of customers) under different priority regimes. Assume that arrivalintensities are given by λ = λ = 1 / λ = 1 / − ε , and priorities b = 3, b = 2 and b = 1.In the case of accumulating priorities one can see (Fig. (1)) that the numbers of customersin all classes become large. If static priorities are used (Fig. (2)), only the number ofcustomers of class 3 grows large, whereas the numbers of customers in classes 1 and 2remain reasonably small at all times. We can also see from Fig. (3) below that if our Class1
Class2
Class3
Figure 2: Numbers of customers of different classes against time in the system with staticpriorities, with ε = 0 . Class1
Class2
Class3
Figure 3: Numbers of customers of different classes against time in the system with accumulatingbut where initial priorities are assigned according to our suggested solution, with ε = 0 . proposed solution is applied, the sample path looks similar to that of the system with staticpriorities.A further example (see figure 4) illustrates how by changing the priority regime one canavoid long waiting times for high-priority customers if there is a sudden surge of lower-priority customers. There are, as before, three priority classes, λ = λ = 1 / λ = 1 / − .
3, so the total load ofthe system is 0 .
7, the system is in relatively light traffic and the accumulating priorities areused. All queue lengths are small. In the second quarter of the simulation time the arrivalrate of the low priority customers suddenly jumps (this could represent for instance seasonaleffects) to λ = 1 / − − , so the total load in the system is 1 − − and the system isin heavy traffic. One can observe all queues becoming large, including that of the highestpriority customers. At this point we switch to the static priority regime and apply it forthe remainder of the simulation time (arrival rates remain such that the system is in heavytraffic). One can observe that the static priority regime ensures that only the low-priorityqueue is large, queues of higher-priority customers are small. In this section we assume that Λ = λ + . . . + λ N < ρ = Λ + λ N +1 > ρ − λ i < i (i.e. the Class 3Class 2Class 1
Figure 4: Switching from accumulating to static priorities stabilises queues of high-prioritycustomers system without any class would be stable). We have two particular cases in mind but donot restrict our attention to these. The first case is similar to the one we had in mind whenstudying the system in heavy traffic: an increase in the lowest-class arrival rate takes thesystem load above capacity. Another case may illustrate a catastrophic event, such as forinstance a pandemic where a sudden jump in the highest-priority patients may lead to asystem operating above capacity.Since ρ >
1, and the system is therefore unstable, in this section we study a fluid versionof the model in which we consider separately the queue for each customer class.We suppose that the level of queue i at time t > L i ( t ), is given by L i ( t ) = L i (0) + λ i t − (cid:90) t D i ( s ) ds, where D i ( s ) denotes instantaneous service rate enjoyed by queue i . In the case of (SP) D ( s ) = I ( L ( s ) >
0) + λ I ( L ( s ) = 0) for s >
0. As long as L , thequeue for class 1 is strictly positive, all the available service capacity is directed to class 1.Once L has emptied, the new arrivals of class 1 are assigned a dedicated service rate of λ .Since λ ≤ Λ <
1, this guarantees that L ( t ) = 0 for all t ≥ T for some finite T .If N ≥
2, then D ( t ) = 0 for t < T . For values of t > T , we have D ( t ) = (1 − λ ) I ( L ( t ) >
0) + λ I ( L ( t ) = 0), that is, a fraction λ of the available rate is used to keep L at zero, and the remaining service capacity is all assigned to queue 2, while it is positive.Once L drops to 0, only a fraction λ of the available capacity is required to drain the queuelength at the same rate at which arrivals occur. Thus, since λ + λ ≤ Λ < L ( t ) = 0 for t > T for some finite T .Similarly to the above, we can conclude that there exists a finite T = T N such that L ( t ) = . . . = L N ( t ) = 0 for t ≥ T . For class N + 1, on the other hand, D N +1 ( t ) = 0 for t < T . When t ≥ T , D N +1 ( t ) = (1 − Λ) I ( L N +1 ( t ) > L (cid:48) N +1 ( t ) = λ N +1 − (1 − Λ) = ρ − >
0, and L N +1 ( t ) → ∞ as t → ∞ . .2 Accumulating priorities When considering accumulating priorities we define the maximal priority process for eachqueue i , 1 ≤ i ≤ N + 1, as P i ( t ) = b i L i ( t ) λ i . Here we have replaced W i ( t ) by L i ( t ) /λ i , which is the age of the oldest fluid particles in thesystem – for the fluid model considered here, these are equivalent.If arg max j { P j ( t ) } is unique, then D i ( t ) = I ( i = arg max j { P j ( t ) } ) , ≤ i ≤ N + 1 . On the other hand, if arg max j { P j ( t ) } is not unique, let J = { i : i ∈ arg max j { P j ( t ) } . Under the accumulating priority regime, if two or more classes have priority max j { P j ( t ) } then service capacity should be divided between them in such a way that their prioritiesremain equal (and maximal). Therefore, if |J | >
1, then P (cid:48) i ( t ) = c , say, for some constant c ∈ R + for all i ∈ J . Thus c = P (cid:48) i ( t ) = b i λ i ( λ i − D i ( t ))and hence D i ( t ) = λ i − λ i b i c for all i ∈ J . But (cid:80) i ∈J D i ( t ) = 1 and hence c = (cid:80) i ∈J λ i − (cid:80) i ∈J λ i /b i . Recall that c = P (cid:48) i ( t ) for all i ∈ J , and recall that we assume ρ − λ i < i . Thus P (cid:48) i ( t ) < i ∈ J as long as J does not consist of the entire set { , . . . , N + 1 } and P (cid:48) i ( t ) > J = { , . . . , N + 1 } .Thus we can now understand the dynamics of the process of priorities: if we start attime 0 with a unique class with the highest fluid priority, its priority is decreasing until itequalises with the priority of another class. From that point onwards, the two priorities staythe same, and both are decreasing at the same rate, until they equalise with the priority ofa further class. This continues until all priorities equalise, from which point onwards thesepriorities grow infinitely. This of course implies that the levels of fluids grow infinitely.Note also that the above may be summarised for the level of queue i as follows: once allpriorities have equalised, L (cid:48) i ( t ) = λ i b i P (cid:48) i ( t ) = λ i b i c i = ( ρ − λ i /b iN +1 (cid:80) j =1 λ j /b j , or lim t →∞ L i ( t ) t = ( ρ − λ i /b iN +1 (cid:80) j =1 λ j /b j , which shows that the relative queue lengths are exactly as in (5). As before, a solution to the possibility of queues growing without bound if the accumulatingpriority regime is applied to all classes is to either employ a static priority regime, or amixture of accumulating and static priority regimes, but in either case the lowest priorityclass needs to be operating under the static priority regime. Both the pure static priorityregime, and the mixture, yield identical fluid solutions for classes 1 , , . . . , N as t → ∞ , with L i ( t ) = 0, t > T S for some T S >
0. On the other hand, L N +1 → ∞ as t → ∞ , under any ofthe regimes. Patients’ strategic behaviour
The analysis throughout the previous sections shows the implications of an exogenous growthof demand that loads the system to its (almost) full capacity. As a consequence, dependingon the priority regime used, some, or all of the class-specific queues may grow without abound.Such an exogenous growth, in the healthcare context, may be a result of a deceaseoutbreak, seasonality, natural disaster, etc. Our findings suggest that the impact of suchincrease in demand may spread over the whole system. What the analysis so far lacks tocapture are secondary effects due to patients behaviour, that is, how patients of differenturgency classes adapt to this change. We leave the exact analysis of such an effect to futureresearch. Nevertheless, we provide some insights on this case.There is a vast literature on strategic customer behavior in queues. For a comprehensiveliterature review see [8] (until 2003) and [7] (from 2004). The most relevant model to thepurpose of our analysis is where customers face a “to queue or not to queue” situation (see,e.g., [6]) where customers decide whether or not to join an unobservable queue while takinginto account that other customers face the same dilemma. Specifically, assume that eachcustomer incurs a cost of
C >
R >
0. In the healthcare context, C may be related to the urgency of the patientand R may be seen as the cost and risk associated with no treatment or the price of analternative (private) treatment that comes with a negligible wait. Here as well, R is relatedto the urgency of the patient condition where relatively low values of R are associated withelective treatments whereas life threatening condition are associated with extremely highvalues.Given a joining strategy of the other customers, a customer evaluates her expected (sta-tionary) waiting time, henceforth denoted by W . Under Nash equilibrium, all the customersuse their best-response strategy against the strategy used by the other customers, that is,they decide to join the queue if CW < R , to balk if
CW > R , and join with any probabilityif CW = R . Note that depending on the model under consideration, C , R , and W may becustomer specific.In the basic model, customers arrive according to a Poisson process at rate λ , servicetimes are exponential with rate µ , the service regime is first-come first-served (FCFS),and customers are homogeneous with respect to their economic parameters C and R . Asymmetric (mixed) strategy here is to join the M/M/1 queue with probability 0 ≤ p ≤ p , the resulting stationary expected waiting timeat the resulting M/M/1 queue is 1 µ − pλ if µ > pλ , and ∞ otherwise. If p is such that C/ ( µ − pλ ) < R , then an individual customeris better off by joining the queue (with probability 1). Likewise, if C/ ( µ − pλ ) > R , arational customer balks. An individual customer is indifferent between the two options if R = C/ ( µ − pλ ). Therefore, a symmetric Nash equilibrium, in which no customer has theincentive to deviate from the common strategy used by the other customers, is a joiningprobability p e such that p e = R > C/ ( µ − λ ) , µ > λ R < C/µp ∗ otherwise , where p ∗ is the unique solution in p of R = Cµ − pλ . The analysis above suggests that if R is high enough, that is R > C/µ , the equilibriumeffective arrival rate as a function of λ equals p e λ = (cid:26) λ λ < µ − C/Rµ − C/R λ ≥ µ − C/R, nd the corresponding equilibrium expected waiting time equals W e = (cid:26) / ( µ − λ ) λ < µ − C/RR/C λ ≥ µ − C/R. (6)In words, for low arrival rates, the system behaves as an ordinary M/M/1 queue, thatis, an increased demand affects expected waiting times. However, past some point, anadditional increase of demand will drive the expected waiting past the service value R andsome customers will therefore prefer the alternative over joining the queue. The resultingeffective arrival rate and expected waiting time are therefore unchanged.Equation (6) also suggests that when the potential demand is high enough, the waitingtimes are in fact determined by the economic parameters R and C . In particular, as R inthe healthcare context is typically extremely high—especially in life-threatening situationsand expensive elective procedures—so are the expected waiting times. Another way to lookat this would be that the existence of private alternatives at a reasonable price may act asa safety valve: the private sector absorbs the public system’s over-demand when waitingtimes are going too high.This basic idea applies also to the multi-class model and priority queues considered in thispaper, where the condition and urgency of the patients not only determines their priorityclass, but also determines their values for R and C . The analysis in Section 3 suggeststhat when the arrival rate grows towards system capacity, the expected waiting time of thelowest priority class (class N + 1) grows without a bound. Regarding the rest of the priorityclasses, their waiting times also grow without a bound in the case of AP whereas they remainfinite under SP. Taking into account that patients react strategically leads to the followingconclusion. If there exist an alternative to class N + 1 patients at a reasonable price R ,their effective arrival rate will stop growing at some point, and their, as well as all the otherclasses’ waiting times will remain finite, even when AP are used. Thus, by creating such analternative (e.g., in the form of reasonably priced private treatment), the effect of high loadsare moderated. For more on equilibrium behaviour in multi-class priority queues, see [9]. We have seen that in heavy traffic, the highest priority classes need greater protection thanis afforded by the accumulating priority queue with fixed accumulation rates. This can beachieved either by permitting accumulation rates to grow in inverse proportion to (cid:15) in thecase ρ = 1 − (cid:15) , or by applying a static priority regime to the lowest priority class. In eithercase the lowest priority class suffers from increasing waiting times, but higher priority classesare protected from this growth.These results have implications for other scenarios where prioritisation of tasks is afeature. We discuss below two other important areas of application, but we believe that thepotential applications are considerably wider.Prioritisation of tasks has been introduced in models of human dynamics where, uponcompleting a task, a person chooses the task from their to-do list with the highest priorityto be performed next. A variant of static priorities has been considered in [2] and a versionof accumulating priorities - in [3]. Few people would disagree with the observation that atleast at some points in our lives we all experience an overload of our to-do lists. This maybe modelled as the arrival rate being (perhaps temporarily) close to, or even above, thecompletion rate, exactly the settings considered in this paper. Our results can therefore beinterpreted as follows: when the number of tasks on the to-do list grows, if time-dependentpriorities are used, the number of outstanding high-importance tasks will grow. In order toprevent this, either static priorities, or a combination of accumulating and static prioritiessuggested here, should be used.Another connection we would like to highlight is to wireless transmission protocols,namely the celebrated MaxWeight introduced in [16]. A simple version of it may be describedas follows: there are a number of queues, each with its own exogenous stream of arrivingjobs, and a single server which, upon completing a job, chooses the next one to performfrom the queue with the largest number of outstanding jobs. Other priorities have also beendiscussed, in particular weighted queue lengths. If one views our model as tasks from the ame class forming a queue, then in the case of accumulating priorities the server choosesthe next task from the queue with the highest weighted waiting time of the longest-waitingtask. Situations considered in this paper are such that the numbers of outstanding tasks inall queues grow to infinity. In this case, the waiting time of the longest-waiting customer isproportional to the number of outstanding tasks. 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