The Average Waiting Time for Both Classes in a Delayed Accumulating Priority Queue Blair Bilodeau 1 and David Stanford 2 1 University of Toronto, Department of Statistical Sciences 2 Western University, Department of Statistical and Actuarial Sciences May 27, 2019 Presented to the Canadian Operational Research Society in Saskatoon, Canada Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 1 / 16
Overview Accumulating Priority Queue 1 Delayed Accumulating Priority Queue 2 Class-2 M/M/1 Waiting Time 3 Class-2 M/M/1 Average Waiting Time 4 Class-1 M/M/1 Average Waiting Time 5 Numerical Examples 6 M/M/c and M/G/1 Extension 7 Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 2 / 16
Accumulating Priority Queue Problem Formulation Class-1 and Class-2 customers arrive with zero priority. Arrival rates: λ 1 , λ 2 ∈ [0 , ∞ ) Priority accumulation rates: b 1 > b 2 ∈ (0 , ∞ ) Service rate: µ ∈ (0 , ∞ ) Stability: ρ := λ 1 + λ 2 < 1 µ Accumulated Priority Consider the n th customer, of class i ( n ) , who arrived at τ n : V n ( t ) = b i ( n ) ( t − τ n ) Limitation Heavily penalizes Class-1 compared to Non-Preemptive Priority Queue. Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 3 / 16
Accumulating Priority Queue A sample of accumulated priority in an Accumulating Priority Queue: V n ( t ) t 0 2 4 6 8 10 12 14 16 18 Class-1 Class-2 Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 4 / 16
Delayed Accumulating Priority Queue Motivation In hospital settings, some patients may not need to be seen urgently until after some time has passed. This allows more preference to be given to Class-1 customers, while not ignoring Class-2. Additional Structure For simplicity, b 1 = 1 and b 2 := b ∈ (0 , 1) Class-2 waits for d ∈ (0 , ∞ ) units of time before accumulating priority Accumulated Priority Consider the n th customer, of class i ( n ) , who arrived at τ n : � t − τ n if i ( n ) = 1 V n ( t ) = b ( t − d − τ n ) if i ( n ) = 2 Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 5 / 16
Delayed Accumulating Priority Queue A sample of accumulated priority in a Delayed APQ: V n ( t ) t 0 2 4 6 8 10 12 14 16 18 Class-1 Class-2 Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 6 / 16
Waiting Time Less Than d Let W DAP,i and W NP,i denote the stationary waiting time for Class- i patients from the Delayed APQ and Non-Preemptive Priority Queue respectively. Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 7 / 16
Waiting Time Less Than d Let W DAP,i and W NP,i denote the stationary waiting time for Class- i patients from the Delayed APQ and Non-Preemptive Priority Queue respectively. Theorem 3.1. (Mojalal et al. 2019) Up to time d , the waiting time for a Class-2 customer in the Delayed APQ is the same as the waiting time for a Class-2 customer in the Non-Preemptive priority queue. That is, P ( W DAP, 2 ≤ t ) = P ( W NP, 2 ≤ t ) ∀ t ∈ [0 , d ] . Implication Since the distribution of W NP, 2 is known, we only have to consider the case when waiting is longer than d units of time. Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 7 / 16
Waiting Time Greater Than d Strategy Consider a Class-2 customer of interest, denoted by X , who has been waiting for d units of time. The following determine the waiting time of X beyond d : 1) The customer currently in service must finish service. 2) All customers in the system at time d with greater priority than X must be served. 3) All customers who accumulate more priority than X before X enters service must be served. These are referred to as accrediting customers. Each customer generates an accreditation interval consisting of their service time plus the service times of all those who accredit during their service. Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 8 / 16
Accreditation Intervals Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 9 / 16
Accreditation Intervals Length of an Accreditation Interval The distribution of the length of an accreditation interval is completely determined by the rate at which customers accredit. Delayed APQ: λ 1 (1 − b ) Non-Preemptive Priority Queue: λ 1 Intuition The reduced waiting time experienced by Class-2 customers in the Delayed APQ can be completely explained by the lower accreditation rate, and consequently shorter accreditation intervals. Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 9 / 16
Waiting Time Greater Than d Let N t be the number of customers in system t time units after arrival. π i := P ( N 0 = i ) P ij ( d ) := P ( N d = j, N t > 0; t ∈ [0 , d ] | N 0 = i ) Denote the Laplace-Stieltjes transform of an accreditation interval for queue type Q by η Q ( s ) . Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 10 / 16
Waiting Time Greater Than d Let N t be the number of customers in system t time units after arrival. π i := P ( N 0 = i ) P ij ( d ) := P ( N d = j, N t > 0; t ∈ [0 , d ] | N 0 = i ) Denote the Laplace-Stieltjes transform of an accreditation interval for queue type Q by η Q ( s ) . Theorem 3.2. (Mojalal et al. 2019) In the M/M/1 case, The LST of the waiting time greater than d is ∞ ∞ � � P ij ( d ) e − sd ( η Q ( s )) j , � � e − s W Q, 2 1 {W Q, 2 > d } E = π i i =1 j =1 Q ∈ { DAP, NP } . Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 10 / 16
Class-2 M/M/1 Average Waiting Time Removing the Laplace-Stieltjes Transform E [ W Q, 2 1 {W Q, 2 > d } ] = − d � � � e − s W Q, 2 1 {W Q, 2 > d } � dsE � s =0 ∞ ∞ � � � � d + jη ′ = π i P ij ( d ) Q (0) . i =1 j =1 Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 11 / 16
Class-2 M/M/1 Average Waiting Time Removing the Laplace-Stieltjes Transform E [ W Q, 2 1 {W Q, 2 > d } ] = − d � � � e − s W Q, 2 1 {W Q, 2 > d } � dsE � s =0 ∞ ∞ � � � � d + jη ′ = π i P ij ( d ) Q (0) . i =1 j =1 Equivalence with Non-Preemptive Priority E [ W DAP, 2 ] = E [ W DAP, 2 1 {W DAP, 2 ≤ d } ] + E [ W DAP, 2 1 {W DAP, 2 > d } ] E [ W NP, 2 ] = E [ W NP, 2 1 {W NP, 2 ≤ d } ] + E [ W NP, 2 1 {W NP, 2 > d } ] Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 11 / 16
Class-2 M/M/1 Average Waiting Time Removing the Laplace-Stieltjes Transform E [ W Q, 2 1 {W Q, 2 > d } ] = − d � � � e − s W Q, 2 1 {W Q, 2 > d } � dsE � s =0 ∞ ∞ � � � � d + jη ′ = π i P ij ( d ) Q (0) . i =1 j =1 Equivalence with Non-Preemptive Priority E [ W DAP, 2 ] = E [ W DAP, 2 1 {W DAP, 2 ≤ d } ] + E [ W DAP, 2 1 {W DAP, 2 > d } ] E [ W NP, 2 ] = E [ W NP, 2 1 {W NP, 2 ≤ d } ] + E [ W NP, 2 1 {W NP, 2 > d } ] Average Waiting Time ∞ ∞ � � � � η ′ NP (0) − η ′ E [ W NP, 2 − W DAP, 2 ] = π i P ij ( d ) j DAP (0) � �� � i =1 j =1 ∆ 2 Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 11 / 16
Class-1 M/M/1 Average Waiting Time Result E [ W NP, 2 − W DAP, 2 ] � ∞ k � e − νd ( νd ) k 1 � � γ ( k ) + ρe − (1 − r )( νd ) = ∆ 2 (1 − ρ ) 1 − ρ + rνd j k ! j =1 k =0 Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 12 / 16
Class-1 M/M/1 Average Waiting Time Result E [ W NP, 2 − W DAP, 2 ] � ∞ k � e − νd ( νd ) k 1 � � γ ( k ) + ρe − (1 − r )( νd ) = ∆ 2 (1 − ρ ) 1 − ρ + rνd j k ! j =1 k =0 Non-Preemptive Priority λ E [ W NP, 2 ] = µ 2 (1 − ρ 1 )(1 − ρ ) Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 12 / 16
Class-1 M/M/1 Average Waiting Time Result E [ W NP, 2 − W DAP, 2 ] � ∞ k � e − νd ( νd ) k 1 � � γ ( k ) + ρe − (1 − r )( νd ) = ∆ 2 (1 − ρ ) 1 − ρ + rνd j k ! j =1 k =0 Non-Preemptive Priority λ E [ W NP, 2 ] = µ 2 (1 − ρ 1 )(1 − ρ ) Conservation Law ρ 2 µ − λ = ρ 1 E [ W DAP, 1 ] + ρ 2 E [ W DAP, 2 ] Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 12 / 16
Effect of Accumulation Rate Class 1, ρ =0.6 2.0 APQ Delayed APQ, d=1 Delayed APQ, d=2 Delayed APQ, d=3 Delayed APQ, d=4 Non−Preemptive 1.5 Expected Waiting Time 1.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 b Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 13 / 16
Effect of Delay Length Class 1, ρ =0.8 5 First Come First Served Delayed APQ, b=1.0 Delayed APQ, b=0.8 Delayed APQ, b=0.6 4 Delayed APQ, b=0.4 Delayed APQ, b=0.2 Non−Preemptive Expected Waiting Time 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 d Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 14 / 16
M/M/c and M/G/1 Extension Multiple Servers When all servers are busy, the queue is indistinguishable from an M/M/1 delayed APQ with service at rate cµ . The probability of all servers being busy is the Erlang-C probability, and can be readily computed. Blair Bilodeau and David Stanford CORS 2019 May 27, 2019 15 / 16
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