The prioritising exclusion process

Transcription

1 The prioritising exclusion process Jan de Gier University of Melbourne GGI workshop, Firenze 2014 With Caley Finn Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

2 Motivation The accumulating priority queue (APQ, Kleinrock 1964) Queueing Syst (2014) 77: DOI /s Waiting time distributions in the accumulating priority queue David A. Stanford Peter Taylor Ilze Ziedins Received: 31 December 2011 / Revised: 30 July 2013 / Published online: 7 December 2013 The Author(s) This article is published with open access at Springerlink.com Abstract We are interested in queues in which customers of different classes arrive to a service facility, and where performance targets are specified for each class. The manager of such a queue has the task of implementing a queueing discipline that results in the performance targets for all classes being met simultaneously. For the case where the performance targets are specified in terms of ratios of mean waiting times, as long Jan de Gier (University of ago Melbourne) as the 1960s, Kleinrock The suggested prioritising a queueing exclusiondiscipline process to ensure that the targets 17 June / 27

3 APQ The accumulating priority queue (APQ): M/M/1 queue (arrival and service rate) with two classes of customers High and low priority customers (triage in emergency) Customers deterministically acquire priority High priority do so at a faster rate Customers are ordered according to their priority Hence they overtake deterministically Change accumulation rates to achieve service waiting times performance levels Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

4 Waiting time performance levels Queueing Syst (2014) 77: Table 1 CTAS key performance indicators Category Classification Access Performance level (%) 1 Resuscitation Immediate 98 2 Emergency 15 min 95 3 Urgent 30 min 90 4 Less urgent 60 min 85 5 Not urgent 120 min 80 of patients whose waiting times before accessing treatment should not exceed the stipulated standard. For example, as is depicted in Table 1, the Canadian Triage and Acuity Scale (CTAS) [4] formulates five priority classifications for assessment in an emergency department, each with its own time standard and compliance target for the proportion of that class s patients that need to meet that standard. The Australasian Triage Scale [3], on which CTAS is based, likewise, has five priority classes, but with different compliance targets. Elective patients awaiting surgery or treatment are also categorized into priorities with compliance targets; we cite as particular examples hip and knee replacement priority scoring in Canada [2] and New Zealand [7], and coronary artery bypass graft surgery in New Zealand [15]. Curtis et al. [6] gavean overview of prioritisation in Australia, as well as a discussion of the Clinical Priority Assessment Criteria (CPAC) tools used in New Zealand and the Western Canada Waiting List Project (WCWL) in Canada. A variant of the accumulating priority mechanism has been considered previously by healthcare modellers in a simulation of emergency care. Hay et al. [9] proposed Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

5 Goals APQ has fluctuating length Stochastic scheduling mechanism? Mapping to exclusion process Goals: Stat-phys: Find a solvable particle hopping model with fluctuating length Queueing: What can we say using physics methods Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

6 Prioritising exclusion The prioritising exclusion process (PEP) λ 1 p μ λ 2 n Low and high arrivals with rate λ 1 and λ 2 High overtakes low with rate p Service rate µ Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

7 M/M/1 queue The PEP is related to the simple M/M/1 queue. The total arrival rate of customers to the PEP is λ = λ 1 + λ 2, and the service rate is µ. Both these rates are independent of the internal arrangement of the queue, and the prioritising parameter p. If we are interested only in the total length of the queue, we can treat the system as a M/M/1 queue with arrival rate λ and service rate µ. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

8 M/M/1 queue The state of a M/M/1 queue is characterised by the probability distribution P n of a queue of lenght n: d P 0 = µp 1 λp 0 d t d P n = λp n 1 + µp n+1 (µ + λ)p n, n > 0. d t The stationary length distribution of the M/M/1 queue (and hence for the PEP) is ( P n = 1 λ ) ( ) n λ, µ µ when λ < µ (arrival rate less than service rate) The expected queue length is finite, given by n = λ µ λ. We will call this the bounded phase of the PEP. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

9 M/M/1 queue At p = 0, the PEP reduces exactly to a M/M/1 queue. Customers are high priority with probability λ 1 /λ or low priority with probability λ 2 /λ. But as there is no overtaking, the probability of high or low at any site is the same as at arrival. For p = 0, in the bounded phase, P(τ n... τ 1 ) = P n ( λ1 λ ) h ( ) l λ2. λ where h = n τ i, l = n h. i=1 Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

10 Unbounded phase When λ > µ, the system is unstable and the expected queue length grows as n (λ µ)t. In the late time limit, we can treat the queue as infinite in length. We call this the unbounded phase of the PEP Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

11 Notation Specify both the site, i, and the lattice length, n: τ i n = P(queue length is n, and site i is occupied). Correlation functions, number from right to left: τ in τ i2... τ i1 n, n i n >... i 2 > i 1 1. Examples of stationary rate equations are 0 = λ 1 P 0 + µ τ 2 2 (λ + µ) τ 1 1, 0 = λ τ 1 n 1 + µ τ 2 n+1 + p τ 2 (1 τ 1 ) n (λ + µ) τ 1 n, n > 1. These couple between different lenghts. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

12 Unbounded queue; Reference frames For any fixed configuration τ n... τ 1, lim t P(τn... τ 1) = 0, The service frame is fixed at the right hand end of the lattice: P serv(τ m... τ 1 ; n) = P(τ n... τ m+1 τ m... τ 1 ). τ n,...,τ m+1 =0,1 The arrival frame probability is defined by P arr(τ 1... τ m; n) = τ m+1,...,τ n=0,1 P(τ 1... τ mτ m+1... τ n). In the t limit, the expected lattice length is infinite, and thus we are interested in the n limits P serv(τ m... τ 1 ) = lim n P serv(τ m... τ 1 ; n), P arr(τ 1... τ m) = lim n P arr(τ 1... τ m; n). Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

13 Domain wall ansatz Consider a general but finite section of length m, with a length k jam: τ m... τ k+2 01 k = τ 01 k. Assume that the conditional probability for a high at site i given a jam of length k is 1 1 i k P(τ i = 1 k) = 0 i = k + 1 α i k + 2, P(τ i =1 n,k) 1 α n k+1 k Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

14 Domain wall ansatz The probability of the finite segment τ = τ m... τ 1 is P serv(τ 01 k ) = P(... τ m+1 τ m... τ k+2 01 k ) where h = τ,...,τ m+1 =0,1 = α h (1 α) l P jam(k). m τ i, l = m h k 1, i=k+2 and P jam(k) is the probability of a length k jam. The jam probabilities are normalised such that P jam(k) = 1. k=0 Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

15 Service frame The stationary rate equation for the k-jam configuration with k 1 is ( 0 = µp serv τ 01 k+1) ) ) + µp serv (τ 01 k 0 + pτ mp serv (0τ 01 k (m+1,m) + m i=k+2 p(1 τ i )τ i 1 P serv ( τ 01 k (i,i 1) ) + pp serv ( τ 101 k 1) µp serv(τ 01 k ) m pτ i (1 τ i 1 )P serv(τ 01 k ) p(1 τ m)p serv(1τ 01 k ). i=k+2 Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

16 Service frame Substituting the domain wall ansatz, the terms representing hopping combine and telescope to ( m ) m p (1 τ i )τ i 1 τ i (1 τ i 1 ) α h (1 α) l P jam(k) i=k+2 i=k+2 = pτ mα h (1 α) l P jam(k). The factor α h (1 α) l is common to all terms in the rate equation. Cancelling, and simplifying leaves 0 = pαp jam(k 1) + µp jam(k + 1) + µ(1 α)α k P jam(0) (µ + pα)p jam(k). No approximation has been made. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

17 Service frame With boundary condition this implies the recursion with solution The normalisation condition fixes P jam(k) = pα µ Pjam(k 1) + αk P jam(0), k 1. ( ) k p pα µ µα k P jam(k) = P jam(0). p µ P jam(0) = (1 α)(1 pα µ ), subject to the constraint No growing jam. pα < µ. What is α? Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

18 Arrival frame The value of α can be determined by considering the arrival frame. One obtains: 0 = λα(1 α) + pα 2 (1 α) pτ 1 α(1 α) + τ 1 λ 1 (1 α) + (1 τ 1 )λ 2 α, which for both τ 1 = 0 and τ 1 = 1 reduces to pα 2 (p + λ)α + λ 1 = 0. The physical solution is the smallest root of this quadratic. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

19 Density profile The density at site i in the service frame, τ i, is computed from the domain wall solution as τ i = α i 2 k=0 Pjam(k) + k=i Pjam(k) ) i = α + (1 α). ( pα µ Τ i i Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

20 Conserved current The bulk equations can be written where d d t τ i = J (i+1) J, (i) J (1) = µ τ 1, J (i) = µ τ i + p τ i (1 τ i 1 ). Stationary current: J = pα(1 α) + µα Low priority customers leave the queue at rate µ J = (µ pα)(1 α). For pα > µ, the jam becomes unbounded, and low priority customers will no longer be served. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

21 Bounded phase λ < µ In the bounded phase the domain wall ansatz breaks down at the arrival end. Define the probability of a length k jam in a length n queue P(n, k) = P(τ n... τ k+2 01 k ). (1) τ n,...,τ k+2 =0,1 Approximation Length assumption: P(n, k) = P np jam(k). The equations for P jam(k) have the same form as for P jam(k) in the unbounded case but with λ in place of µ. Pjam(k) = p ( ) pα k λα k λ Pjam(0). p λ Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

22 Density profile ( pα ) ) i τ i n = P n (α + (1 α). λ Τ i npn Τ i npn i (c) pα < λ i (d) pα > λ Triangle markers show simulation results, with points for each length connected by dashed lines. Domain wall approximation invalid for pα > λ. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

23 Aggregate density profile and current Another approximation can be derived for aggregate densities: τ i = τ i n. n=i where with stationary solution d d t τ i = J (i) ext + J(i+1,i) J (i,i 1), J (i,i 1) = µ τ i + p τ i (1 τ i 1 ). J (i,i 1) = λ 1 ( λ µ ) i 1. Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

24 Aggregate density profile τ i = ( λ µ ) i [ α + (1 α) ( pα µ ) i ], i Τ in i i Τ in i i Τ in i i Τ in i Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

25 Waiting times We can use the aggregated densities to compute the average number of customers in the queue Define N 1 = τ i n = τ i. Average waiting time of high priority customers is W 1 = 1 N 1 = 1 ( α λ ) λ 1 λ 1 µ λ + (1 α) pαλ. µ 2 pαλ i=1 n=i i= W i (i) Low density p W i (j) High density Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27 p

26 Conclusions Conclusion Queuing model with stochastic scheduling Fluctuating length Domain wall ansatz exact for the unbounded phase Two approximations for bounded phase Waiting times Outlook Full exact solution More classes of particles Waiting time distributions Fluctuations and large deviations? Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27

27 Papers Exlusion processes: arxiv: Exclusion in a priority queue Jan de Gier and Caley Finn arxiv: , J. Phys. A 44 (2011) Relaxation rate of the reverse biased asymmetric exclusion process Jan de Gier, Caley Finn and Mark Sorrell Traffic modelling: arxiv: , Physica A 401 (2014), Traffic disruption and recovery in road networks Lele Zhang, Timothy M. Garoni and Jan de Gier arxiv: , Transport. Res. B: Meth. 49 (2013), 1-23 A comparative study of Macroscopic Fundamental Diagrams of arterial road networks governed by adaptive traffic signal systems Lele Zhang, Timothy M. Garoni and Jan de Gier Jan de Gier (University of Melbourne) The prioritising exclusion process 17 June / 27