1. No category

Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
On Waiting Times for Nonlinear Accumulating
Priority Queues
David A. Stanford
Department of Statistical and Actuarial Sciences
University of Western Ontario
(Co-authors: Na Li, Peter Taylor, Ilze Ziedins)
April 10, 2015
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
1 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Outline
1
Introduction
2
Our Model
3
Power-Law APQs
4
Nonlinear APQs
5
Numerical examples
6
Future Work
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
2 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Motivation
Table: CTAS Key Performance Indicators (KPIs)
Level
1
2
3
4
5
Level of acuity
Resuscitation
Emergent
Urgent
Less urgent
Non urgent
Response time
Immediate
< 15 mins
< 30 mins
< 60 mins
< 120 mins
Diagnosis
Cardiac arrest
Chest pain
Moderate asthma
Minor trauma
Common cold
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
Targets
98%
95%
90%
85%
80%
April 10, 2015
3 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Linear Accumulating Priority Queue [Stanford et al.]
Customers accumulate priority linearly over time: higher
priority, greater rate.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
4 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Linear Accumulating Priority Queue [Stanford et al.]
Customers accumulate priority linearly over time: higher
priority, greater rate.
Class-1
Class-2
=⇒
Service Node
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
4 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Linear Accumulating Priority Queue [Stanford et al.]
Priorities accumulated linearly: higher priority, greater rate.
Class-1
Class-2
=⇒
Service Node
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
4 / 21
Introduction
Our Model
Power-Law APQs
Class-1
Nonlinear APQs
Numerical examples
Future Work
References
Class-2
=⇒
Service Node
Priority
t1
t2
time
(a) Linear APQ
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
4 / 21
Introduction
Our Model
Power-Law APQs
Class-1
Nonlinear APQs
Numerical examples
Future Work
References
Class-2
=⇒
Priority
Service Node
Priority
t1
t2
time
(a) Linear APQ
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
t1
t2
time
(b) Nonlinear APQ
April 10, 2015
4 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Our Model
A multi-class accumulating priority queue with Poisson arrivals
and general service time distributions.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
5 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Our Model
A multi-class accumulating priority queue with Poisson arrivals
and general service time distributions.
A nonlinear priority accumulation discipline, defined by a set
F of priority accumulation functions {fk (.), k = 1, ..., K }.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
5 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Our Model
A multi-class accumulating priority queue with Poisson arrivals
and general service time distributions.
A nonlinear priority accumulation discipline, defined by a set
F of priority accumulation functions {fk (.), k = 1, ..., K }.
Specifically a customer from class k who arrived at time t0
has priority, at time t > t0 ,
qk (t) = fk (t − t0 ).
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
(1)
April 10, 2015
5 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Our Model
A multi-class accumulating priority queue with Poisson arrivals
and general service time distributions.
A nonlinear priority accumulation discipline, defined by a set
F of priority accumulation functions {fk (.), k = 1, ..., K }.
Specifically a customer from class k who arrived at time t0
has priority, at time t > t0 ,
qk (t) = fk (t − t0 ).
(1)
The next customer to be served will be the customer with the
greatest priority at a service completion instant.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
5 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Our Model
The functions in F have the properties, for k = 1, ..., K :
I
I
I
I
fk is a strictly increasing, differentiable function that maps to
R+ to R+ .
fk (0) = fk+1 (0), WLOG, fk (0) = 0.
for j < k and for every t ∈ R+ , fj (t) > fk (t).
If there exists a t ∗ such that fj (t ∗ − t0 ) = fk (t ∗ ), then
fj0 (t ∗ − t0 ) > fk0 (t ∗ ).
Remark: This implies that there is at most one crossing point.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
6 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Our Model
The functions in F have the properties, for k = 1, ..., K :
I
I
I
I
fk is a strictly increasing, differentiable function that maps to
R+ to R+ .
fk (0) = fk+1 (0), WLOG, fk (0) = 0.
for j < k and for every t ∈ R+ , fj (t) > fk (t).
If there exists a t ∗ such that fj (t ∗ − t0 ) = fk (t ∗ ), then
fj0 (t ∗ − t0 ) > fk0 (t ∗ ).
Remark: This implies that there is at most one crossing point.
qk (t)
qj (t) = fj (t − t1 )
qk (t) = fk (t − t2 )
t2
t1
t∗
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
t
April 10, 2015
6 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
A Case with No Crossing Point
7
6
5
4
3
Series1
2
Series2
Series3
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
-1
-2
-3
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
7 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Waiting Times for Power-Law APQ I
Kleinrock and Finkelstein (1967) studied the expected waiting times
in a multi-class single-server queue with exponential service times,
where the priority accumulates as a power function of the incurred
waiting time.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
8 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Waiting Times for Power-Law APQ I
Kleinrock and Finkelstein (1967) studied the expected waiting times
in a multi-class single-server queue with exponential service times,
where the priority accumulates as a power function of the incurred
waiting time.
Accumulation functions for the power-law APQ of order r :
For a sequence {bk , k = 1, ..., K }, such that, 1 > b1 > · · · > bK > 0,
fk (t) = bk t r .
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
(2)
April 10, 2015
8 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Waiting Times for Power-Law APQ II
Kleinrock and Finkelstein [4, Theorem 1]:
If one were to set
1/r 0
1/r 0
(r )
(r )
(r )
(r 0 )
bk+1 /bk
= bk+1 /bk
, k = 1, ..., K ,
(3)
then the expected waiting times of all classes of customers in the
corresponding power-law APQs of orders r and r 0 are identical.
By setting r 0 = 1 we can compare the given nonlinear system to a linear
one. Since the ranking of customers is the same at all points in time, we
refer to it as the “linear proxy” of the nonlinear system. We obtain
1/r
bkL = bk where bkL denotes the corresponding coefficient for class k in
the linear proxy.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
9 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Waiting Times for Power-Law APQ III
Lemma
Consider a power law APQ of order r and its linear proxy, both
starting empty and driven by the same relations of the arrival and
service time processes. Then for any time t ∈ R, the priority
ordering in the system of order r , Γ(r ) (t), is the same as the
ordering Γ(t) in the linear proxy; i.e. Γ(r ) (t) = Γ(t).
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
10 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Waiting Times for Power-Law APQ IV
Theorem
Consider a single server APQ which has a set F of priority accumulation
(r )
functions fk (t) = bk t r for k = 1, 2, . . . , K and a positive parameter r .
Pk
Let δk = j=1 ρj 1 − (bk+1 /bj )1/r . The LST of the delayed waiting
˜ +(k) (s; r ), is given by
time distribution for a class-k customer, W
(k)
˜ (k+1) ((bk+1 /bk )1/r s; r )
˜ (k) (s; r ) = 1 − (bk+1 /bk )1/r W
˜ acc
W
(s; r ) + (bk+1 /bk )1/r W
+
+
(4)
where ...
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
11 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Waiting Times for Power-Law APQ III
Theorem
... where
(k)
˜ acc
˜ (k+1) ((bk+1 /bk )1/r s; r )
W
(s; r ) = W
+
+
K
X
j=k+1
k
X
ρj (bk+1 /bk )1/r ˜ (k,j)
Wacc (s; r )
1 − δk
j=1
(5)
ρj
1 − ρ ˜ (k,0)
(k,j)
˜ (j) ((bj /bk )1/r s; r )W
˜ acc
W
(s; r ) +
Wacc (s; r )
1 − δk +
1 − δk
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
11 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Question
Do there exist other nonlinear systems with linear proxies than the
power-law functions?
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
12 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
A Sample Realization of the Linear APQ
12
accumï
ulated
priority
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
time
class 1, b1 = 1
class 2, b2 = b = 0.5
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
13 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Nonlinear APQ I
Lemma
Suppose a nonlinear APQ has a set F of priority accumulation functions fk (t)
for k = 1, ..., K . If the ratio
c(k) (t) =
0
fk+1
(t)
,
0 −1
fk (fk (fk+1 (t)))
0 < c(k) < 1 for all t > 0.
(6)
is a constant c(k) (t) = c(k) with respect to t, then there exists a corresponding
equivalent linear system.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
14 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Nonlinear APQ II
Theorem
For any nonlinear APQ, assuming f1 (t) is known, if there exists a
sequence of constants c(k) , 0 < c(k) < 1 ∀k = 1, 2, . . . , K − 1 such that
fk+1 (t) = fk (c(k) t),
t > 0.
(7)
then an equivalent linear system exists.
Therefore, once we have made a selection for the highest priority
function, sat, f1 (t) = g (t), then the remaining functions must
come front the same family, but the time scale is changed due to
the product of elements in the sequence {c(k) }.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
15 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Some Examples of the Nonlinear Accumulation Functions
Power-law functions: for k = 1, 2, . . . , K ,
1/r
qk (t) = bk (t − t0 )r = [bk (t − t0 )]r ,
1/r
qkL (t) = bkL (t − t0 ) = bk (t − t0 )
1/r
1/r
where g (x) = x r , ⇒ c(k) = bk+1 /bk .
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
16 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Some Examples of the Nonlinear Accumulation Functions
Power-law functions: for k = 1, 2, . . . , K ,
1/r
qk (t) = bk (t − t0 )r = [bk (t − t0 )]r ,
1/r
qkL (t) = bkL (t − t0 ) = bk (t − t0 )
1/r
1/r
where g (x) = x r , ⇒ c(k) = bk+1 /bk .
Exponential functions: for k = 1, 2, . . . , K ,
qk (t) = exp(bk (t − t0 )),
where g (x) = exp(x), ⇒ c(k) = bk+1 /bk .
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
16 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Some Examples of the Nonlinear Accumulation Functions
Power-law functions: for k = 1, 2, . . . , K ,
1/r
qk (t) = bk (t − t0 )r = [bk (t − t0 )]r ,
1/r
qkL (t) = bkL (t − t0 ) = bk (t − t0 )
1/r
1/r
where g (x) = x r , ⇒ c(k) = bk+1 /bk .
Exponential functions: for k = 1, 2, . . . , K ,
qk (t) = exp(bk (t − t0 )),
where g (x) = exp(x), ⇒ c(k) = bk+1 /bk .
Similarly, for logarithm functions and generalized exponential
functions.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
16 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Common Factors in the Numerical Examples
Two class of patients (CTAS-3 & CTAS-4): λ1 = 1 and
λ2 = 0.75.
Single server. The service time for both classes are
exponentially distributed at a common rate µ1 = µ2 = 2.
Power-law APQ: b1 ≡ 1 and b2 = b = 0.5, r = 1/3, 1, 3.
Gaver-Stehfest numerical inversion algorithm.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
17 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Numerical Examples
(a)
(b)
Figure: Waiting Time Distributions for the Power-Law APQ
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
18 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Future Work
1
Finding the waiting time distributions for the linear APQ with
affine functions as its accumulation functions.
2
Deriving the waiting time distributions for the nonlinear APQ
for which an equivalent linear system does not exist.
3
Optimization problems for the nonlinear APQ.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
19 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
References
Abate J., Whitt W., A unified framework for numerically inverting Laplace
transforms, Institute for Operations Research and Management Sciences
(INFORMS) Journal on Computing, 18:408-421, 2006.
Canadian Association of Emergency Physicians. The Canadian Triage and Acuity
Scale. http://www.cfhi-fcass.ca/migrated/pdf/chartbook/CHARTBOOK%
20Eng_June_withdate.pdf.
Kleinrock L., A delay dependent queue discipline, Naval Research Logistics
Quarterly 11. pp.329-341, 1964.
Kleinrock L. and Finkelstein R., Time dependent priority queues, Operations
Research, Vol. 15, No.1, 1967, pp. 104-116.
Kleinrock L., Queueing systems Vol I & II, Wiley, New York, 1975 & 1976.
Stanford D.A., Taylor P., Ziedins I., Waiting time distributions in the
accumulating priority queue, Queuing System. 2014.
Sharif A.B., Stanford D.A., Taylor P., Ziedins I., A multi-class multi-server
accumulating priority queue with application to health care, Operations Research
for Health Care. 2014.
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
20 / 21
Introduction
Our Model
Power-Law APQs
Nonlinear APQs
Numerical examples
Future Work
References
Thank you for your attention!
On Waiting Times for Nonlinear Accumulating Priority Queues - David A. Stanford
April 10, 2015
21 / 21