Strategic Choice of Arrival Times
in a Transient Queue
Nahum Shimkin, Technion
Joint work with
Sandeep Juneja, TIFR
Workshop on Optimization, Scheduling and Queues
In Honor of Gideon Weiss
The University of Haifa, June 2012
The Arrival-Time Problem
We consider a (FCFS) queueing system that caters to a finite
population of customers
Service starts at a known time T0
Customers prefer to get served early (or before others), but
dislike waiting in queue
The customer's dilemma: When to arrive at the queue?
• arrive early and (possibly) wait longer in queue,
or
• arrive later: wait less but finish late
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Some Examples
Lining up …
for tickets: e.g. for a football match or rock concert
(early buyers get better seats)
for a new product (new iPad, Harry Potter book)
for lunch in a busy cafeteria
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The Queueing Model
N customers, FCFS
I.I.D. service demands (V j , j = 1,… , N ) , exponentially distributed
with E (V j ) = µ −1
Service starts at T0 = 0
Each customer j can choose her arrival time t j ∈ (−∞, ∞) ,
possibly randomly, according to a CDF F j (t )
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Cost Function
We consider a linear cost function for each customer:
C j = α j w j + β jτ j
where
• w j is j 's waiting time in the queue
• τ j his service completion time
Homogeneous population case: α j ≡ α , β j ≡ β
Each customer wishes to minimize his own expected cost E (C j ) ,
by appropriately choosing his arrival time distribution F j (t )
Due to obvious dependence on arrival decision of others – we
obtain a non-cooperative game problem, and consider the Nash
Equilibrium Point (NEP) for this game.
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Related Literature
Strategic arrivals in queues:
Glazer & Hassin (1983): exponential system with opening time,
no late-service cost.
Similar models: Rapoport et. al. (2004), Hassin & Kleiner (2009),
Haviv (2012)
Lariviere & van Mieghem (2004): competition leads to Poisson
arrivals.
Transportation literature:
Extensive work exists on the bottleneck model (Vickrey 1969),
also known as the morning commute problem. This essentially
comprises a fluid queueing model, with individual costs related to
required arrival times.
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The Fluid Model
Motivated by the large-population scenario ( N → ∞ ), we first consider
a simplified fluid model of our stochastic queueing system.
Consider a continuous population of (infinitesimal) customers,
represented by the set [0, Λ ].
Deterministic service rate
µ , starting at t = 0 .
Thus, with no idleness, all customers may be served within T f = Λ
µ
Suppose there are I ≥ 1 customer classes, indexed by i , with
respective masses {Λ i }, ∑ i Λ i = 1.
The cost function for a class- i customer is
Ci ( w,τ ) = α i w + βiτ
( w : waiting time in queue, τ : service completion time
α i > 0, βi > 0 : class-dependent cost parameters)
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Fluid model: Equilibrium arrival profile
Define mi =
αi
α i + βi
, and suppose mi < mi +1
The equilibrium arrival density is then unique and has the following form
(Jain-Juneja-S. 2011)
Λ
F '(t )
3
µ m3
class 3
µ m3
class 1
Q (t )
µ m2
µ m1
T0
T1
T2
T3 = Λµ
t
Evidently: Those who do not mind waiting (smaller mi ) arrive earlier.
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Fluid Model: Single-Class
Suppose I = 1 (homogeneous cost functions)
Then the equilibrium arrival
density is uniform:
F '(t )
µ αα+ β
T0 = − αβ T1
−α
0
T1 =
cost C ( t )
β
t
Λ
µ
t
The Price of Anarchy (PoA, ratio of social cost at equilibrium to
optimal social cost) is exactly 2
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Fluid Model: Single-Class Case (2)
Note: While the aggregate arrival profile is unique, it may be
composed in many different ways from individual arrival
densities. Hence, there is no uniqueness on the individual level.
aggregate
t
individual
t
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Back to the Stochastic Queueing Model
We return to the N -customer queueing model, with
homogeneous costs (i.e., single-class).
Main results:
♦ Existence and uniqueness of a (symmetric) NEP
♦ Characterization and computation of the NEP
♦ Convergence of to the fluid limit as N → ∞
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Notation
Strategy profile: F = {F j }Nj =1,
F −i = {F j , j ≠ i}
Q(t ) ≡ Q(t ; F ) , Q −i (t ) ≡ Q(t ; F −i )
Ci (t ) = E (α wi (t ) + βτ i (t )) : expected cost for i given that he
arrives at t and the others follow F −i
Queue size:
As τ i (t ) = t + wi (t ) , we have
Ci (t ) = (α + β ) E ( wi (t )) + β t
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Equilibrium Conditions
Let T i denote the support of (the measure represented by) Fi (⋅)
Then, at a Nash equilibrium,
Ci (t ) = ci for t ∈ Ti , and Ci (t ) ≥ ci for t ∉ Ti
Therefore, on the interior of T i ,
d C (t )
dt i
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Equilibrium Conditions
We next observe that
Ci (t ) = (α + β ) E ( wi (t )) + β t
= (α + β )(
d
dt
E (Q −i (t )) =
E (Q −i (t ))
µ
d
dt
+ max{0, −t}) + β t
E ( A−i (t ) − S −i (t ))
= ∑ j ≠i F j' (t ) − µ1{t ≥0} P (Q −i (t ) > 0))
d C (t ) = 0, t ∈ T o is equivalent to
So that dt
i
i
1
µ
'
F
∑ j ≠i j (t ) =
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α
α +β
− 1{t ≥0} P(Q −i (t ) = 0) ,
Strategic Choice of Arrival Times
t ∈ Ti o
14
Main Results: Symmetry
Proposition 1
Any NEP is symmetric, namely Fi = F ∀i
[Established under mild technical conditions on { F }]
i
We next characterize the symmetric equilibrium distribution F .
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Existence, Uniqueness and Characterization
Proposition 2
1. F (t ) admits a density F '(t ) , and is supported on a single interval
F '( t )
[ta , tb ], with ta < 0 and tb > 0
µ
2. For ta ≤ t < 0 , F '(t ) = N −1 αα+ β , a constant
ta
tb
0
3. For 0 ≤ t < tb , F (t ) satisfies the differential relation
N −1 F '(t )
µ
= αα+ β − P{Q −i (t ; F ) = 0}
(*)
−i
Here Q (t ; F ) is the queue size due to arrivals of the (other)
N − 1 customers with arrival-time distribution F
4. F (tb ) = 1 (by definition), and F '(tb ) = 0
5. Properties 1-4 uniquely characterize the equilibrium dist. F
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t
Note:
Existence of solutions to the (functional) differential equation (*)
follows by Lipschitz continuity of the RHS in F (tb ) = 1.
Existence and uniqueness of the solution with the prescribed
boundary conditions follows by monotonicity properties of the
RHS.
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Random N
Suppose that N , the number of arriving customers, is random
2
with known distribution p N = ( p N ( n), n ≥ 1) , with E ( N ) < ∞ .
Then the results above are remain valid, with the deterministic
N replaced by
2
E
(
N
)
N E( N )
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Computation
Explicit computation of the equilibrium distribution F is possible for
N = 2 only. In that case, F '(t ) is linear for 0 ≤ t ≤ tb .
For N > 2 , a numerical solution is required.
♦
♦
P{Q −i (t ; F ) = 0} may be computed, for given F , from the evolution
equations for the Markov chain ( k (t ), m(t )) , where k is the number in
the queue and m the number of arrival by time t , 0 ≤ k ≤ m < N . These
equations may be integrated together with (*) to obtain F from any initial
value F (0) .
A search over F (0) is required to find the solution that satisfies the
boundary condition.
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Computation results
The following equilibrium arrival densities
F '(t ) are obtained for α = 2 ,
β = 1, µ = N , for several values of N .
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The Fluid Limit
Consider the N -th system, with N customers, service rate
µ N = µ N , and equilibrium distribution FN .
Recall that for the fluid model with volume Λ = 1 and rate µ , we
β
had f (t ) ≡ F '(t ) = µ αα+ β for t ∈ [Ta , Tb ] = [− µα , µ1 ] .
Proposition 3: F N (t ) converges to F (t ) at a rate o(
log N
),
N
in
the sense that:
taN = Ta − o(
N −1
N
log N
)
N
tbN = Tb + o(
log N
)
N
f N (t ) = µ αα+ β − µ P0 (t )1{t ≥0} , t ∈ [taN , tbN ]
log N
where 0 ≤ P0 (t ) ≤ N1 for t ≤ Tb − o(
)
N
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Further Work
Heterogeneous cost parameters
Nonlinear costs, due-date arrival times.
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