Existence and uniqueness of a fluid model for many-server queues with abandonment

Operations Research Letters 42 (2014) 478–483

Contents lists available at ScienceDirect

Operations Research Letters journal homepage: www.elsevier.com/locate/orl

Existence and uniqueness of a fluid model for many-server queues with abandonment Weining Kang Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21250, United States

article

info

Article history: Received 18 April 2014 Received in revised form 11 August 2014 Accepted 13 August 2014 Available online 23 August 2014

abstract In this paper, we establish a new (direct) proof for existence and uniqueness of the fluid model for Gt /GI /N + GI queues proposed recently by Kang and Ramanan under mild conditions on the arrival, service and patience time distributions. In particular, the existence of the fluid model is established directly from the fluid model itself. Then main technique here is the application of two non-linear functional integral equations, one of which is of Volterra type. © 2014 Elsevier B.V. All rights reserved.

Keywords: Gt /GI /N + GI queue Fluid model Existence and uniqueness Non-linear functional integral equations Volterra type

1. Introduction In recent years, there has been tremendous attention to the study of many-server queueing systems with customer abandonment due to its applications to telephone contact centers and (more generally) customer contact centers; see, e.g., [2–5], and references therein. In this paper, we consider a many-server queueing system, also known as Gt /GI /N + GI model, where there are N parallel identical servers, and customers arrive with a (possibly) timedependent arrival rate, require i.i.d. service times, and have i.i.d. patience times. Customers are assumed to abandon from the system if the time spent waiting in queue reaches their patience time. The arrival process, service and patience times are assumed to be mutually independent. The service discipline is first-come–firstserve (FCFS) and non-idling, that is, no server will idle whenever there is a customer in queue. For such a system, the exact analysis is intractable and several fluid models [6–8,11,12] have recently been developed to approximate the system dynamics in the, so-called, many-server heavy-traffic regime, where the arrival rate and the number of servers get large, while service and patience time distributions are fixed. To justify these fluid models as the approximations to the Gt /GI /N + GI system under the fluid scaling in the many-server heavy-traffic regime, one of the key steps is to establish existence and uniqueness of these fluid models. In this paper, we focus on the measure-valued fluid model proposed in Kang and Ramanan [7], since it is quite general without

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.orl.2014.08.006 0167-6377/© 2014 Elsevier B.V. All rights reserved.

any restrictions on the customers’ arrival pattern before time zero. The existence of this fluid model was established indirectly by first showing a sequence of fluid scaled measure-valued state descriptors for the Gt /GI /N + GI queue is tight and then verifying any weak limit of the sequence satisfies the fluid model. Viewing this fluid model as a non-linear deterministic dynamical system, we provide a new (direct) proof for the existence and uniqueness of this system under general initial conditions and mild conditions on the service and patience time distributions. In particular, the existence of the fluid model is established directly from the fluid model itself. The main technique in the proof is the use of two non-linear functional integral equations, one of which is of the Volterra type. We believe that the technique used here is quite general and can be adapted to establish existence and uniqueness of other dynamical systems with similar structure. In addition, the new (direct) proof enables us to establish a new algorithm to compute various timedependent performance quantities in the fluid model by choosing suitable numerical schemes to solve the two non-linear functional integral equations and the complexity and convergence rate analysis of this new algorithm can also be conducted using properties of the two non-linear functional integral equations. This will be carried out in future study. 2. Fluid model In this section we first state the fluid model for Gt /GI /N + GI queues proposed in [7]. Then we state the main result on the existence and uniqueness of the fluid model under the following assumption.

W. Kang / Operations Research Letters 42 (2014) 478–483

479

Assumption 2.1. The arrival process E is absolutely continuous, the service time distribution Gs has density g s on its support [0, H s ) and the patience time distribution Gr has density g r on its support [0, H r ) and its hazard rate function hr = g r /(1 − Gr ) is a.e. locally bounded.

We refer the reader to [7] (right after Definition 3.3 of [7]) for an informal, intuitive explanation of the fluid model equations in Definition 2.2. From the definition of the fluid model equations, we have the following two additional balance equations: for each t ≥ 0, from (2.5) and (2.9),

Remark 2.1. Assumption 2.1 holds if E has arrival rate function t λ(·), that is, E (t ) = 0 λ(s)ds for each t ≥ 0, the service time distribution Gs is absolutely continuous, and the patience time distribution Gr is locally Lipschitz continuous.

Q (t ) = X (t ) − 1

The following notation is needed to state the definition of the fluid model. Let IR+ denote the set of non-decreasing, càdlàg functions on [0, ∞) with f (0) = 0, R+ denote the set of non-negative real numbers, MF [0, H s ) denote the space of finite measures on [0, H s ), MFc [0, H r ) denote the space of finite, continuous measures on [0, H r ), [x]+ denote the maximum of x and 0, Cb (R+ ) denote the space of bounded, continuous functions on R+ , and for any Borel measurable function f : [0, H ) → R that is integrable with respect to measure ξ on [0, H ), we often use the short-hand notation .  . ⟨f , ξ ⟩ = [0,H ) f (x) ξ (dx). For a c.d.f G, let G¯ = 1 − G. Define the following space of feasible input data for the fluid model equations,

  . (e, x, ν, η) ∈ IR+ × R+ × MF [0, H s ) × MFc [0, H r ) : S0 = . + 1 − ⟨1, ν⟩ = [1 − x] , x ≤ ⟨1, ν⟩ + ⟨1, η⟩



t



t



⟨hr , ηu ⟩ du < ∞, 0

⟨hs , ν u ⟩ du < ∞,

(2.1)

0

and the following relations are satisfied: for every f ∈ Cb (R+ ),

⟨f , ν t ⟩ =

 [0,H s )

f (x + t )

¯ s (x + t ) G ¯ s ( x) G

ν 0 (dx)

t



¯ s (t − u) dL(u), f (t − u)G

+

(2.2)

0

where L(t ) = ⟨1, ν t ⟩ − ⟨1, ν 0 ⟩ +



t

⟨hs , ν u ⟩ du;

(2.3)

0

⟨f , ηt ⟩ =

 [0,H r )

f (x + t )

¯ r (x + t ) G ¯ r ( x) G

t



η0 (dx)

¯ r (t − u) dE (u); f ( t − u) G

+

(2.4)

0

Q (t ) = X (t ) − ⟨1, ν t ⟩ ≤ ⟨1, ηt ⟩; R(t ) =

 t  0

[0,H r )

 11[0,χ(s)] (u)hr (u)ηs (du) ds,

(2.5)

where

  χ(s) = inf x > 0 : ηs [0, x] ≥ Q (s) ;  t X (t ) = X (0) + E (t ) − ⟨hs , ν u ⟩ du − R(t );

(2.7) (2.8)

0

and the non-idling condition

+

1 − ⟨1, ν t ⟩ = 1 − X (t )



.

(2.9)

(2.10)

Q (0) + E (t ) = Q (t ) + L(t ) + R(t ).

(2.11)

We now state the main result of the paper. Theorem 2.3. Suppose that Assumption 2.1 holds. The fluid model equations in Definition 2.2 admit a unique solution. 3. Proof of Theorem 2.3 We start with some derivations. Define the following two functions A and B as follows. For each t , y ≥ 0, define

.

A(t , y) =

[y−t ]+



¯ r (x + t ) G ¯ r (x) G

0

η0 (dx) +



η0 (dx) +



t

[t −y]+

¯ r (t − u)dE (u), G (3.12)

.

B(t , y) =

[y−t ]+



g (x + t ) r

¯ r (x) G

t

[t −y]+

g r (t − u)dE (u). (3.13)

Since the integrands in (3.12) and (3.13) are non-negative, η0 is a non-negative continuous measure and E is continuous and nondecreasing, then for each t ≥ 0, A(t , ·) and B(t , ·) are non-negative, non-decreasing and continuous and we denote them alternatively as At (·) and Bt (·), respectively. Thus, the inverse functions of At (·) 1 and Bt (·) are well defined and denoted, respectively, by A− and t . −1 −1 Bt , where for each x ≥ 0, At (x) = inf{y ≥ 0 : At (y) ≥ x} and . 1 B− t (x) = inf{y ≥ 0 : Bt (y) ≥ x}, with the convention that inf ∅ 1 + = ∞. Moreover, it is clear that At (A− t (x)) = [x] ∧ At (∞) and −1 + Bt (Bt (x)) = [x] ∧ Bt (∞), where a ∧ b denotes the minimum of a and b. 1 Lemma 3.1. For each t ≥ 0, Bt (A− t (·)) is non-negative, non-decreasing and continuous on R. 1 Proof. Fix t ≥ 0. Note that the function Bt (A− t (·)) is non-negative 1 and non-decreasing since both Bt and A− are non-negative and t 1 non-decreasing. We next show that Bt (A− (·)) is continuous on R. t 1 It is clear that Bt (A− (·)) is continuous on ( A (∞), ∞) and (−∞, t t 1 −1 0) since Bt (A− ( x )) = B (∞) when x > A (∞) and B t t t (At (x)) = 0 t when x < 0. For each x ∈ (At (t ), At (∞)), if At (·) does not have 1 a constant stretch at x, then Bt (A− t (·)) is right continuous at x. On 1 the other hand, if At (·) has a constant stretch at x, then A− t (x) is the −1 left end and At (x+) is the right end of the stretch. Since x > At (t ), 1 −1 −1 then A− t (x+) > At (x) > t. By (3.12), 0 = A(t , At (x+)) − A(t , 1 A− t (x)) =

 A−t 1 (x+)−t

−1 At (x)−t −1 At x

¯ r (x+t ) G ¯ r (x) G

η0 (dx). Since

¯ r (x+t ) G ¯ r (x) G 1 t A− t

is positive on

(x+) − t ] = 0. ( +) − t ], then η0 [At (x) − , This, combining the definition of B in (3.13), implies that Bt (·) 1 −1 −1 −1 is constant on [A− t (x), At (x+)] and then Bt (At (x)) = Bt (At −1 (x+)). Thus, Bt (At (·)) is right continuous at x. It is obvious that 1 Bt (A− t (·)) is left continuous at x by the definitions of A and B, then it is continuous on (At (t ), At (∞)). Similarly, we can also show that 1 Bt (A− t (·)) is continuous on (0, At (t )). It is clear that limx↑At (∞) −1 1 −1 Bt (At (x)) = Bt (∞) = Bt (A− t (At (∞))), limx↓0 Bt (At (x)) = 0, −1 −1 limx↓At (t ) Bt (At (x)) = Bt (t ) and limx↑At (t ) Bt (At (x)) = Bt (t ), 1 then Bt (A− t (·)) is continuous at 0, At (t ) and At (∞). This establishes the lemma.  [At (x) − t , −1

(2.6)

,

and from (2.3), (2.5) and (2.8),

0

Definition 2.2. The càdlàg function (X , ν, η) defined on R+ such that X ∈ R+ , ν ∈ MF [0, H s ), and η ∈ MFc [0, H r ) is said to solve the fluid model equations associated with (E , X (0), ν 0 , η0 ) ∈ S0 and the hazard rate functions hr and hs if and only if for every t ∈ [0, ∞),

+

−1

480

W. Kang / Operations Research Letters 42 (2014) 478–483

Let L denote the unique solution to the following renewal equation f (t ) =

Gs (x + t ) − Gs (x)



¯ s (x) G

[0,H s )

ν 0 (dx)

t



g (t − u)f (u) du. s

+

(3.14)

0

Since the first term on the right-hand side of (3.14) is continuous and bounded, by the key renewal theorem (see, e.g., Theorem 4.3 in Chapter V of [1]), L admits the following representation L(t ) =

Gs (x + t ) − Gs (x)



¯ s (x) G

[0,H s )

+

¯ s ( x) G

[0,H s )

0

ν 0 (dx)

Gs (x + t − w) − Gs (x)

 t 

 ν 0 (dx) us (w) dw, (3.15)

where us is the density of the renewal function U s associated with Gs . (Note that us exists because Gs is assumed to have a density.) Obviously, L is continuous and non-decreasing on [0, ∞) with L(0) = 0. Moreover, it is clear from (3.15) that L is absolutely continuous  ′ ′ g s (x+t ) and its a.e. derivative L is given by L (t ) = [0,H s ) G¯ s (x) ν 0 (dx) +

 t  0

g s (x+w) [0,H s ) G¯ s (x)

 ν 0 (dx) us (t − w)dw.

Now consider the following functional integral equation: f (t ) +

t



1 + Bu (A− u (f (u))) du = [X (0) − 1] + E (t ) − L(t ). (3.16)

0

Lemma 3.2. Eq. (3.16) admits a unique solution on [0, ∞) and the unique solution is continuous on [0, ∞). Proof. For the uniqueness, let f1 and f2 be two solutions to (3.16). t 1 −1 Note that by (3.16), f1 (t ) − f2 (t ) = 0 (Bu (A− u (f2 (u))) − Bu (Au (f1 (u))))du. Thus, it follows that 0 ≤ (f1 (t ) − f2 (t ))2 = 2

t



(f1 (u) − f2 (u))d(f1 (u) − f2 (u)) 0

t



1 −1 (f1 (u) − f2 (u))(Bu (A− u (f2 (u))) − Bu (Au (f1 (u))))du.

=2 0

1 Since, for each u ≥ 0, Bu (A− u (·)) is non-decreasing by Lemma 3.1, it is clear that the integrand in the above display is non-positive, and hence f1 = f2 . For the existence, we shall apply the existence results developed in [10] and extended in [9]. Our Eq. (3.16) corresponds to their integral equation in one dimension with W = R, I = [0, ∞), 1 f (t ) = [X (0)− 1]+ + E (t )− L(t ), a(t , u) = 1, g (x, u) = Bu (A− u (x)). + It is clear that [X (0)− 1] + E (t )− L(·) is continuous on [0, ∞) and takes values in R, then Hypothesis A in [10] and [9] holds. It is clear 1 −1 that Bt (A− t (x)) as a function of (x, t ) is measurable and Bt (At (x)) is continuous in x for each t ≥ 0 by Lemma 3.1. Moreover, note 1 that Bt (A− t (x)) ≤ Bt (∞) for each x ∈ R and t ∈ [0, ∞) and for each compact interval J ⊂ [0, ∞),



Bt (∞)dt ≤ J

∞



g r (x + t )



¯ (x)

Gr

[0,H r )

0

N



g (t − u)dE (u)dt r

+ 0

Now let Y denote the unique solution to (3.16). It is clear that Y (0) = [X (0) − 1]+ . 1 Lemma 3.3. For each t ∈ [0, ∞), Y (t ) ≤ At (∞) and then At (A− t (Y (t ))) = Y (t ).

Proof. Since E and L are absolutely continuous, then by (3.16), Y ′ is also absolutely continuous and its a.e. derivative Y is given by ′ ′ ′ −1 Y (t ) = E (t ) − L (t ) − Bt (At (Y (t ))). Note that At (∞) as a function in t is also absolutely continuous with a.e. derivative given by ′ At (∞)′ = E (t ) − Bt (∞). This shows that a.e., dtd (Y (t ) − At (∞)) = ′

1 −L (t )− Bt (A− t (Y (t )))+ Bt (∞). So it is clear that for a.e. t such that 1 d Y (t ) > At (∞), −Bt (A− t (Y (t ))) + Bt (∞) = 0 and then dt (Y (t ) − At (∞)) ≤ 0. This shows that Y (t )−At (∞) ≤ 0 for each t ∈ [0, ∞).

The second part of the lemma follows directly from the definition 1 of A−  t . Define

  . σ = inf t ≥ 0 : Y (t ) = 0 .

Proof. If σ = 0, then the result of the lemma holds automatically. Now assume that σ > 0. Then Y (t ) > 0 for each t ∈ [0, σ ) by the definition of σ . Since Y (0) = [X (0) − 1]+ , it follows that ⟨1, ν 0 ⟩ = 1. Define η from the data (E , η0 ) using the right-hand side of . (2.4). For each t ∈ [0, σ ), define χ , Q , R  in sequence by χ (t ) =

.

≤ η0 [0, H r ) +

[0,H r )

11[0,χ (s)] (u)hr (u)

ηs (du) ds. Since L is non-decreasing, define ν from the data (L, ν 0 ) using the right-hand side of (2.2), then it is clear that ν t ∈ MF [0, . H s ) for each t ∈ [0, σ ). At last, for each t ∈ [0, σ ), define X (t ) = Q (t ) + ⟨1, ν t ⟩. We next show that the constructed triple (X , ν, η) with the associated processes Q , R, L and χ is a solution to the fluid model equations (2.1)–(2.9) on [0, σ ). It is clear from the above definitions that (2.2) and (2.4)–(2.6) hold. It follows from the definitions of η, Q and χ (t ), (3.12), and Lemma 3.3 that for each t ∈ [0, σ ), Q (t ) = At (χ (t )) = Y (t ). This implies that χ satisfies (2.7). To show that (2.3) holds, note that from the definition of ν , we have t



⟨h , ν u ⟩du = s

0

Gs (x + t ) − Gs (x)

 [0,H s )

v

 t

+ 

0

¯ s (x) G

t

ν 0 (dx)

g s (v − u) dL(u)dv

G (x + t ) − Gs (x)

[0,H s )



¯ s (x) G

0 s

=

ν 0 (dx)

L(u)g s (t − u) du < ∞,

+

(3.18)

0

⟨1, ν t ⟩ =

¯ s (x + t ) G



¯ s (x) G

[0,H s ) t

 + 

Gr (N − u)dE (u) < ∞,

where N is chosen such that J ⊂ [0, N ]. Then Hypothesis Bp in [10] and [9] holds with p = 1. At last, a(t , u) = 1 clearly satisfies Hypothesis Cp in [10] and [9] with p = 1. Then, it follows from parts

0



N 0

. t 

1 A− t (Y (t )), Q (t ) = η t [0, χ (t )], R(t ) =

0



(3.17)

Lemma 3.4. There exists a unique solution (X , ν, η) to the associated fluid model equations (2.1)–(2.9) on [0, σ ). In addition, if σ < ∞, (X , ν, η) can be extended to the unique solution on [0, σ ].

η0 (dx)dt

t



A and B of Theorem 1 of [10] that Eq. (3.16) admits a solution on [0, ∞). At last, it is clear that any solution to (3.16) is continuous since E and L are continuous on [0, ∞). This completes the proof of the lemma. 

ν 0 (dx)

¯ s (t − u) dL(u) G 0

=

¯ s (x + t ) G

[0,H s )

+ L(t ) −

¯ s (x) G

ν 0 (dx)

t



L(u)g s (t − u) du. 0

(3.19)

W. Kang / Operations Research Letters 42 (2014) 478–483

Adding the above two displays and rearranging terms yield (2.3). t By the definition of η, we also have that 0 ⟨hr , ηu ⟩du < ∞. Then it is clear that (2.1) also holds. From (2.3), the definition of X and the fact that L is the unique solution to (3.14), it is clear that (2.8) holds. By (3.18), (2.3) and the fact that L is the unique solution to (3.14), we also have that ⟨1, ν t ⟩ = ⟨1, ν 0 ⟩ = 1. By the definition of X , we have that (2.9) holds. Thus, we established that the constructed (X , ν, η) is a solution to the fluid model equations (2.1)–(2.9) on [0, σ ). To prove that (X , ν, η) is the unique solution on [0, σ ), suppose ∗ that there exists another solution (X , ν ∗ , η∗ ), then it is not hard to ∗ see using (2.8), (2.10) and (2.11) that [X (·) − 1]+ and [X (·) − 1]+ ∗ are two solutions to (3.16) and then [X − 1]+ = [X − 1]+ by the uniqueness of solutions to (3.16) established in Lemma 3.2. Then the proof of the existence result established earlier shows ∗ that we can recover (X , ν, η) from [X − 1]+ and (X , ν ∗ , η∗ ) from ∗ ∗ ∗ ∗ [X − 1]+ and thus (X , ν, η) = (X , ν , η ). At last, if σ < ∞. , (X , ν, η) can be extended to [0, σ ] by defining (X (σ ), ν σ , ησ ) = limt ↑σ (X (t ), ν t , ηt ). This completes the proof of the lemma.  If σ = ∞, Theorem 2.3 follows directly from Lemma 3.4. To complete the proof of Theorem 2.3, we suppose that σ < ∞. In this case, by Lemma 3.4, we have χ (σ ) = 0 and Q (σ ) = 0. Let E

[σ ]

(·) = E (σ +·)− E (σ ), X

[σ ]

] [σ ] (0) = X (σ ) ≤ 1, ν [σ 0 = ν σ , η0 =

ησ . Now we consider the fluid model equations (2.1)–(2.9) with [σ ] [σ ] [σ ] [σ ] the input data (E , X (0), ν 0 , η0 ) ∈ S0 . Since χ (σ ) = 0, the [σ ] mass in η0 will have no impact to solutions to fluid model equa[σ ] [σ ] [σ ] [σ ] tions (2.1)–(2.9) with the input data (E , X (0), ν 0 , η0 ) ∈ S0 . [σ ] Thus, we may assume, without loss of generality, that η0 is a zero measure on [0, H r ). We now define two functions F and H. For each t , y ≥ 0, define . t . t r ¯ r (t − u)dE [σ ] (u) and H (t , y) = F (t , y) = [t −y]+ G [t −y]+ g (t − u)

481

Proof. Fix T ∈ (0, H r ) and two pairs of functions (x1 , y1 ), (x2 , y2 ) ∈ D [0, ∞) × D [0, ∞) and t ≥ 0. From the definition of Λ in (3.22), it is clear that

|Λ(x1 , y1 )(t ) − Λ(x2 , y2 )(t )|  t  1  x (u) ∧ Fu (∞) − x2 (u) ∧ Fu (∞) g s (t − u)du ≤ 0   t  u   −1 1 −1 2   Hl ((Fl ) (y (l))) − Hl ((Fl ) (y (l))) dl + 0

0

× g s (t − u)du  t   Hu ((Fu )−1 (x1 (u))) − Hu ((Fu )−1 (x2 (u))) du. + 0

Note that x1 (u) ∧ Fu (∞) − x2 (u) ∧ Fu (∞) ≤ x1 (u) − x2 (u)

     1    and y (u) ∧ Fu (∞) − y2 (u) ∧ Fu (∞) ≤ y1 (u) − y2 (u) for each u ∈ [0, t ]. By (3.21), (3.20), we have that   t  u   Hl ((Fl )−1 (y1 (l))) − Hl ((Fl )−1 (y2 (l))) dl g s (t − u)du 0 0   t  u  1  y (l) − y2 (l) dl g s (t − u)du, ≤ CTr 0 0  t   − 1 Hu ((Fu ) (x1 (u))) − Hu ((Fu )−1 (x2 (u))) du 0 t



CTr x1 (u) − x2 (u) du.



≤



0

The lemma is then proved by using the above three displays.



Now, let us consider the following functional integral equation:

  x(t ) = ξ [σ ] (t ) + Λ [x(·) − 1]+ , [x(·) − 1]+ (t ),

(3.20)

(·) is a continuous function determined by  t  ¯ s (x + t ) [σ ] G . [σ ] ¯ r (t − u)dE [σ ] (u). (3.24) ν 0 (dx) + G ξ (t ) = s (x) ¯ s G 0 [0,H ) It is clear from the definition of Λ that any solution (if it exists) to Eq. (3.23) is continuous. It is also clear that ξ [σ ] (t ) > 0 for each t ∈ [0, ∞). Consider the following equation in λ: λ2 − (Gs (T ) + CTr T )λ − CTr TGs (T ) = 0. Let λ1 (T ) and λ2 (T ) be its two roots.

¯ r (x) and hr is a.e. locally bounded, In addition, since g (x) = h (x)G we have that for each T ∈ (0, H r ), there exists a constant CTr > 0 such that for each t ∈ [0, T ] and 0 ≤ a < b < ∞,

Lemma 3.6. For each T ∈ (0, H r ) such that (Gs (T )+CTr T (1+Gs (T ))) < 1, |λ1 (T )| < 1 and |λ2 (T )| < 1, then Eq. (3.23) admits a unique solution on [0, T ].

Ht (b) − Ht (a) ≤ CTr (Ft (b) − Ft (a)).

Proof. Fix T ∈ (0, H r ) such that (Gs (T ) + CTr T (1 + Gs (T ))) < 1, |λ1 (T )| < 1 and |λ2 (T )| < 1. It is clear from Lemma 3.5 that Λ(x, x) as a functional of x is a contraction map on [0, T ] and thus, Eq. (3.23) admits at most one solution. For the existence, let us de-

[σ ]

dE (u). It is clear that for each t ≥ 0, F (t , ·) and H (t , ·) are nondecreasing and continuous and we denote them alternatively by Ft (·) and Ht (·), respectively. Their inverse functions are denoted, respectively, by (Ft )−1 and (Ht )−1 . It is clear that Ft (Ft−1 (x)) = [x]+ ∧ Ft (∞) r

and Ht (Ht−1 (x)) = [x]+ ∧ Ht (∞). r

(3.21)

Let Λ : D [0, ∞)× D [0, ∞) → C [0, ∞) be the following functional map defined by

Λ(x, y)(t )  t  . = x(u) ∧ Fu (∞) + 0



Hl ((Fl )−1 (y(l)))dl g s (t − u)du

−

t

Hu ((Fu )−1 (x(u)))du.

[σ ],−1



The following lemma shows that the map Λ is locally Lipschitz continuous. Lemma 3.5. Let T ∈ (0, H r ). The functional map Λ satisfies the following property: For any two pairs of functions (x1 , y1 ), (x2 , y2 ) ∈ D [0, ∞) × D [0, ∞) and any t ∈ [0, T ],

0≤u≤t

[σ ],0

(3.22)

0

|Λ(x1 , y1 )(t ) − Λ(x2 , y2 )(t )|   ≤ sup x1 (u) − x2 (u) (Gs (t ) + CTr t ) 0≤u≤t   + sup y1 (u) − y2 (u) CTr tGs (t ).

[σ ],n

, n ∈ Z+ } recursively as fol. lows. Let Y =Y = 0 and for each n ∈ N, let  + [σ ],n . [σ ], n −1 Y = ξ [σ ] + Λ Y (·) − 1 , fine a sequence of processes {Y

u 0



where ξ

(3.23)

[σ ]

Y

[σ ],n−2

+  (·) − 1 .

(3.25)

For each n ∈ N, by Lemma 3.5 and the fact that |[x − 1]+ − [y − 1]+ | ≤ |x − y|, we have that

 

sup Y

[σ ],n+1

0≤t ≤T

(t ) − Y

[σ ],n

  (t )

  [σ ],n−1  [σ ],n  ≤ sup Y (t ) − Y (t ) (Gs (T ) + CTr T ) 0≤t ≤T   [σ ],n−2  [σ ],n−1  + sup Y (t ) − Y (t ) CTr TGs (T ). 0≤t ≤T

482

W. Kang / Operations Research Letters 42 (2014) 478–483

A shows that for each n ≥ 0, sup0≤t ≤T  simple induction argument 

[σ ],n−1  [σ ],n  (t ) − Y (t ) ≤ (Gs (T )+ CTr T )λ1 (T )n + CTr TGs (T )λ2 (T )n . Y



t s

[σ ],n

, n ∈ Z+ } is the Cauchy Then it follows that the sequence {Y under the uniform norm on [0, T ]. Combining this and the unique[σ ],n

ness proved earlier, we have that Y converges uniformly on [0, T ] as n → ∞. It is clear that the limit is the unique solution to Eq. (3.23) on [0, T ].  Now, fix a T [σ ] ∈ (0, H r ) such that (Gs (T [σ ] ) + CTr [σ ] T [σ ] (1 + Gs (T [σ ] ))) < 1, |λ1 (T [σ ] )| < 1 and |λ2 (T [σ ] )| < 1. Let Y the unique solution on [0, T [σ ] ] to (3.23) by Lemma 3.6. Lemma 3.7. There exists a constant T [σ ]

[Y

[σ ]

(t ) − 1] ≤ Ft (∞) for each t ∈ [0, T +

[σ ]

[σ ]

denote

∈ [0, T [σ ] ] such that ].

[σ ]

[σ ]

(0) = X (0) = ⟨1, ν 0 ⟩ ≤ 1. It follows that . [Y (0) − 1] = 0 = F0 (∞). For each t ∈ [0, T [σ ] ], let p(t ) = [σ ] [σ ] Y (t ) − 1 − Ft (∞). Since Y is the unique solution on [0, T [σ ] ] to Eq. (3.23), we have that, for each t ∈ [0, T [σ ] ],  t [σ ] p(t ) ≤ φ(t ) + ([Y (u) − 1]+ ∧ Fu (∞) − Fu (∞))g s (t − u)du Proof. Note that Y [σ ]

[σ ]

+

0

≤ φ(t ) +

p(u)g (t − u)du,

.

(3.26)

 s Gs (x) [σ ] (t ) − Ft (∞) − [0,H s ) G (x+G¯ ts)− ν 0 (dx) − ( x )    t [σ ]  [σ ] t u E (u)g s (t − u)du + 0 Hl ((Fl )−1 ([Y (l) − 1]+ ))dl g s (t − 0 0 t t [σ ] u)du − 0 Hu ((Fu )−1 ([Y (u) − 1]+ ))du + 0 Fu (∞)g s (t − u)du. where φ(t ) = E

[σ ]

Note that by applying the integration-by-parts, we have that for each t ∈ [0, T [σ ] ],

0

u

Hl ((Fl )

−1

([Y

[σ ]



(l) − 1] ))dl g s (t − u)du +

t

Hu ((Fu )−1 ([Y

−

[σ ]

(u) − 1]+ ))du

t



¯ s (t − u)Hu ((Fu )−1 ([Y G

[σ ]

(u) − 1]+ ))du ≤ 0.

0

In addition, by changing the order of integration and then applying the integration-by-parts, we have that for each t ∈ [0, T [σ ] ], E

 t [σ ] (t ) − (E (u) − Fu (∞))g s (t − u)du − Ft (∞) 0   t  t [σ ] r r s = G (t − v) − G (u − v)g (t − u)du dE (v) 0 s   t  t [σ ] s r ¯ = G (t − u)g (u − v)du dE (v).

[σ ]



 t 

t

v

0



¯ s (t − u)g r (u − v)du dE G

t



¯ s (t − u)Hu ((Fu )−1 ([Y G

−

[σ ]

.

t

v

[σ ] [σ ] Proof. Define η[σ ] from the data (E , η0 = 0) using the right-

hand side of (2.4). Then for each t ≥ 0, ηt

[σ ]

t ∈ [0, T

[σ ]

[σ ]

[σ ]

∈ MFc [0, H r ). For each

. ] [σ ] (t ) = η[σ (t )], t [0, χ   t  [σ ] . ] R (t ) = 11[0,χ [σ ] (s)] (u)hr (u)η[σ ( du ) ds, s Q

[σ ]

L

[σ ]

[0,H r )

[σ ] [σ ] . [σ ] (t ) = E (t ) − Q (t ) − R (t ).

[σ ] [σ ] At last, define ν [σ ] from the data (L , ν 0 ) using the right-hand

. [σ ] ] (t ) = Q (t ) + ⟨1, ν [σ t ⟩. It follows [σ ] from Lemma 3.7 and the definitions of η[σ ] and F that Q (t ) = [σ ] [σ ] + [Y (t ) − 1] . By the fact that X (0) ≤ 1 and the definitions of [σ ] [σ ] [σ ] [σ ] [σ ] [σ ] L , Q and R , we have that L (t ) = E (t ) − [Y (t ) − 1]+ t [σ ] [σ ] − 0 Hu (Fu )−1 [Y (u) − 1]+ du. We claim that L is nonside of (2.2) and define X

[σ ]

[σ ],n

] ν [σ 0 (dx) +

[σ ],n

, n ≥ 0} as follows. Recall the sequence of functions {Y , n ≥ 0} constructed in the proof of Lemma 3.6 by (3.25). It follows [σ ],n is absolutely from (3.25) that inductively, for each n ≥ 0, Y [σ ],n continuous and then [Y (t ) − 1]+ ∧ Ft (∞) is also absolutely [σ ],0 . [σ ] continuous. Define L (t ) = E (t ) and for each n ≥ 1, det [σ ],n [σ ],n . [σ ] fine L (t ) = E (t ) − [ Y (t ) − 1]+ ∧ Ft (∞) − 0 Hu (Fu )−1  [σ ],n−1  [σ ],n [Y (u) − 1]+ du. Then for each n ≥ 0, L is absolutely [σ ],n

) denote its a.e. derivative. We prove

is non-decreasing for each n ≥ 1. Suppose

[σ ],1

[σ ]

[σ ],1

(t ) = E (t )−[Y (t )− 1]+ ∧ Ft (∞). [σ ] ′ When Y (t ) < 1, (L ) (t ) = (E ) (t ) ≥ 0. When 1 ≤ [σ ],1 [σ ],1 Y (t ) < 1 + Ft (∞), then by (3.24) and (3.25), L (t ) = (1 −   t [σ ] [σ ] Gs (x+t )−Gs (x) [σ ] s X (0))+ [0,H s ) ν ( dx )+ E ( u ) g ( t − u )du. When 0 ¯ s (x) 0 G t r [σ ],1 [σ ],1 [σ ] Y (t ) ≥ 1 + Ft (∞), L (t ) = E (t ) − Ft (∞) = 0 G (t − u) [σ ],1 ′

[σ ]

[σ ],1

(u). Combining the above three cases, it is clear that L is [σ ],i non-decreasing. Suppose for each i < n, L is non-decreasing. [σ ],n [σ ],n [σ ],n ′ [σ ] Now we consider L . When Y (t ) < 1, (L ) (t ) = (E )′ [σ ],n [σ ],n (t ) ≥ 0. When 1 ≤ Y (t ) < 1 + Ft (∞), then by (3.25), L (t )   t [σ ],n−1 [σ ] Gs (x+t )−Gs (x) [σ ] = (1 − X (0)) + [0,H s ) ν ( dx ) + L ( t − u) 0 ¯ s (x) 0 G

(v)

Gs (x+t )−Gs (x) ¯ s (x) G

[σ ]

], define χ [σ ] , Q , R , L in sequence by   [σ ] . ] χ [σ ] (t ) = inf x > 0 : η[σ (t ) − 1]+ , t [0, x] ≥ [Y

[σ ],1

(u) − 1]+ ))du.

Note that the function ϕ(t ) = − [0,H s ) 



[σ ]

, ν [σ ] , η[σ ] ) to the [σ ] associated fluid model equations (2.1)–(2.9) on [0, T ].

first that n = 1. Then L

0



[σ ]

Lemma 3.8. There exists a unique solution (X

dE

[σ ]

g r (t − v) −

(v). Then, there exists a constant

′

by induction that L

Gs (x + t ) − Gs (x) [σ ] ν 0 (dx) ¯ s (x) G [0,H s )

+

0

[σ ],n ′

Combining the above two displays and the definition of φ , we have that, for each t ∈ [0, T [σ ] ],

φ(t ) = −

[σ ]

continuous and then let (L

v

0

 t

∈ [0, T ] such that ϕ (t ) ≤ 0 for a.e. t ∈ [0, T ]. This im[σ ] plies that ϕ(t ) ≤ 0 and then φ(t ) ≤ 0 for each t ∈ [0, T ]. From [σ ] this and (3.26), we have that p(t ) ≤ 0 for each t ∈ [0, T ]. This [σ ] is sufficient to show that [Y (t ) − 1]+ ≤ Ft (∞) for each t ∈ [σ ] [0 , T ].  T

{L

0

=−

[σ ]

[σ ]

[σ ]

decreasing. To establish this, we construct a sequence of functions

0



g s (t − u)g r (u − v)du dE

0

s

0

 t 



t



g s (x+t )

its a.e. derivative ϕ ′ (t ) = − [0,H s ) G¯ s (x) ν 0 (dx) +

t 0

¯ s (t − u)g r (u − v)du dE [σ ] (v) is absolutely continuous and G

g s (u)du  . WhenY

[σ ],n

[σ ],n

[σ ]

(t ) ≥ 1 + Ft (∞) (t ) = E (t )− Ft (∞)−  , L t r [σ ] −1 + H ( F ) [ Y ( u ) − 1 ] du = G (t − u)dE (u) − u u 0 0  [σ ],n−1  t  Hu (Fu )−1 [Y (u) − 1]+ du. In this case, we have that 0

t

[σ ],n−1

W. Kang / Operations Research Letters 42 (2014) 478–483



 [σ ],n−1  [σ ],n ′ [σ ] (L ) (t ) = 0 g r (t −u)dE (u)−Ht (Ft )−1 [Y (t )−1]+ t t t [σ ] [σ ] ≥ 0 g r (t − u)dE (u)− Ht (∞) = 0 g r (t − u)dE (u)− 0 g r (t − t

[σ ]

[σ ],n

u)dE (u) = 0. Combining the three cases, we have that L is non-decreasing. This completes the induction steps. Now, since

{L

[σ ],n

, n ≥ 0} is a sequence of non-decreasing functions and [σ ],n [σ ] [σ ] converges uniformly on [0, T ] to L due to the uni-

clearly L

form convergence of Y

[σ ],n

to Y

[σ ]

[σ ]

by Lemma 3.6. This implies that

L is also non-decreasing on [0, T each t ≥ 0.

[σ ]

]. Then ν t

[σ ]

∈ MF [0, H ) for s

[σ ]

It is clear from the above definitions that (X , ν [σ ] , η[σ ] ) satisfies (2.2) and (2.4)–(2.6). Next, by the definition of η[σ ] , the fact [σ ]

[σ ]

(t ) = [Y (t ) − 1]+ and the definition of χ [σ ] , it is clear that (X , ν [σ ] , η[σ ] ) satisfies (2.7). Using the same argument in Lemma 3.4 that shows that (X , ν, η) constructed in Lemma 3.4 sat[σ ] isfies (2.1), (2.3) and (2.8), we can show that (X , ν [σ ] , η[σ ] ) also [σ ] satisfies (2.1) and (2.3) and (2.8). At last, we show that (X , ν [σ ] , [σ ] η[σ ] ) satisfies (2.9). Recall that Y is the unique solution to Eq. (3.23) on [0, T [σ ] ]. Then by the definition of Λ, Lemma 3.7 and t [σ ] the definition of ξ [σ ] in (3.24), for each t ∈ [0, T [σ ] ], Y (t ) = 0  [σ ]  u [σ ] [Y (u) − 1]+ + 0 Hl ((Fl )−1 ([Y (l) − 1]+ ))dl g s (t − u)du + t  t [σ ] [σ ] ξ [σ ] (t )− 0 Hu ((Fu )−1 ([Y (u)− 1]+ ))du = ξ [σ ] (t )+ 0 (Q (u)+  [σ ] [σ ] [σ ] [σ ] R (u))g s (t − u)du − R (t ) = (X (0) + E (t )) − [0,H s )  t [σ ] [σ ] [σ ] Gs (x+t )−Gs (x) [σ ] ν 0 (dx) − 0 (E (u) − Q (u) − R (u))g s (t − u)du − ¯ s (x) G that Q

[σ ]

[σ ]

[σ ]

R (t ). Then by the definition of L and using the representat ] [σ ] tion of 0 ⟨hs , ν [σ u ⟩du using the definition of ν u , we have that Y

[σ ]

(t ) = X [σ ]

[σ ]

(0) + E

[σ ]

[σ ]

(t ) −

t 0

[σ ]

] ⟨hs , ν [σ (t ). This shows u ⟩du − R [σ ]

= X on [0, T [σ ] ]. Hence, (X , ν [σ ] , η[σ ] ) satisfies (2.9) [σ ] [σ ] [σ ] and then (X , ν [σ ] , η[σ ] ) with its associated processes Q , R , [σ ] L and χ [σ ] is a solution to the fluid model equations (2.1)–(2.9) [σ ] on [0, T ]. that Y

To prove that (X

483

[σ ]

[σ ]

, ν [σ ] , η[σ ] ) is the unique solution on [0, T ], [σ ],∗ , ν [σ ],∗ , η[σ ],∗ ), suppose that there exists another solution (X [σ ]

[σ ],∗

and X are two solutions then it is not hard to see that X [σ ] [σ ],∗ to the integral (3.23) and then X = X by the uniqueness of solutions to (3.23) established in Lemma 3.6. Then the proof of the existence result established earlier shows that we can re[σ ] [σ ],∗ cover (ν [σ ] , η[σ ] ) from X and (ν [σ ],∗ , η[σ ],∗ ) from X and thus

(X

[σ ]

, ν [σ ] , η[σ ] ) = (X

[σ ],∗

, ν [σ ],∗ , η[σ ],∗ ).



Proof of Theorem 2.3. It follows from Lemmas 3.4 and 3.8 that the fluid model equations (2.1)–(2.9) admit a unique solution on

[0, σ + T

[σ ]

). By applying a simple contradiction argument and using Lemmas 3.4 and 3.8, it is clear that the maximal interval on which the fluid model equations (2.1)–(2.9) admit a unique solution has to be [0, ∞). This completes the proof of the theorem.  References [1] S. Asmussen, Applied Probability and Queues, second ed., Springer-Verlag, New York, 2003. [2] L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, L. Zhao, Statistical analysis of a telephone call center: a queueing-science perspective, J. Amer. Statist. Assoc. 100 (469) (2005) 36–50. [3] N. Gans, G. Koole, A. Mandelbaum, Telephone call centers: tutorial, review and research prospects, Manuf. Serv. Oper. Manag. 5 (2003) 79–141. [4] O. Garnett, A. Mandelbaum, M.I. Reiman, Designing a call center with impatient customers, Manuf. Serv. Oper. Manag. 4 (2002) 208–227. [5] L.V. Green, Queueing analysis in healthcare, in: R. Hall (Ed.), Patient Flow: Reducing Delay in Healthcare Delivery, Springer, 2006. [6] W. Kang, G. Pang, Equivalence of fluid models for Gt /GI /N + GI queues (2014) submitted for publication. [7] W. Kang, K. Ramanan, Fluid limits of many-server queues with reneging, Ann. Appl. Probab. 20 (2010) 2204–2260. [8] W. Kang, K. Ramanan, Asymptotic approximations for the stationary distributions of many-server queues, Ann. Appl. Probab. 22 (2012) 477–521. [9] N. Kikuchi, S. Nakagiri, An existence theorem of solutions of non-linear integral equations, Funkcial. Ekvac. 15 (1972) 131–138. [10] R. Miller, G. Sell, Existence, uniqueness and continuity of solutions of integral equations, Ann. Mat. Pura Appl. 80 (1) (1968) 135–152. [11] W. Whitt, Fluid models for multiserver queues with abandonment, Oper. Res. 54 (1) (2006) 37–54. [12] J. Zhang, Fluid models of many-server queues with abandonment, Queueing Syst. 73 (2013) 147–193.