A note on a limit interchange for many-server queues

Operations Research Letters 48 (2020) 147–151

Contents lists available at ScienceDirect

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

A note on a limit interchange for many-server queues Jun Pei a , Amir Motaei b , Petar Momčilović c , a b c

∗

School of Management, Hefei University of Technology, Hefei, China Walmart Labs, San Bruno, CA 94066, United States of America Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843, United States of America

article

info

Article history: Received 29 May 2019 Received in revised form 16 October 2019 Accepted 30 January 2020 Available online 7 February 2020 Keywords: Machine repair QED regime Heavy traffic

a b s t r a c t A connection between open and closed many-server queueing systems is examined. Two limits are considered: (i) the number and reliability of machines (customers) increase simultaneously while the offered load remains constant (Poisson limit), and (ii) the number of machines (customers) and repairmen (servers) increase while the utilization remains close to unity (QED limit). It is argued that the two limits are interchangeable. © 2020 Elsevier B.V. All rights reserved.

1. Introduction The machine repair model is a standard queueing model. It is relevant in describing systems with a finite population of customers that repeatedly return for service. Examples include systems based on membership/subscription [11], professional and warranty services, and computer networks [10]. In the health care context, one can think of patients recovering in a hospital bed and requiring assistance from medical personnel multiple times throughout the day. Outpatient facilities providing dental, oncology and dialysis services can be modeled under this framework as well. Many primary care physicians manage finite panel sizes [5], and thus the machine repair model can be relevant in that setting as well. On the one hand, closed queueing systems might seem more relevant in general because the world is arguably finite [10]. On the other hand, the use of open queueing models is widespread, since closed queueing systems are typically harder to analyze than their open counterparts [13]. The machine repair system consists of a finite number of machines and repairmen. A machine operates for a random amount of time before it breaks down, and it takes a repairman a random amount of time to fix a machine. The broken machines are repaired in the first-come first-served fashion. Once a machine is repaired, it becomes operational, and the cycle repeats. There exists a fundamental difference in the behavior of open and closed systems. In an open system, the arrival rate is independent of the number of customers in the system. In contrast, due to a feedback present in a closed system, the arrival rate decreases as ∗ Corresponding author. E-mail addresses: [email protected] (J. Pei), [email protected] (A. Motaei), [email protected] (P. Momčilović). https://doi.org/10.1016/j.orl.2020.01.010 0167-6377/© 2020 Elsevier B.V. All rights reserved.

the number of customers in the system increases. In general, a closed model might be relevant for systems with the following features: (i) the system operator has a fixed constituency from which service requests materialize, and (ii) once processed, customers return to the pool of potential users and seek additional service later. We focus on a large-scale (many machines and many repairmen) system operating in the Quality-and-Efficiency-Driven (QED) regime. Informally, under such a regime, the probabilities that a machine (upon a breakdown) finds an available repairman and a repairman (upon a repair completion) finds a broken machine are strictly in (0, 1). Analogous to open systems, the capacity of the system is roughly equal to the offered load. Studies of the QED regime are partially motivated by the fact that QED-based approximations adequately describe finite-size systems that operate in efficiency- or quality-driven regimes [10]. The Poisson limit provides a way to model an open system as a limit of closed systems. In particular, the number of repairmen is kept constant, and the number of machines and their reliability is increased so that the offered load remains fixed. We establish a connection between closed [7] and open [8] systems in the QED regime via a Poisson limit [2]. In particular, we show that the two limits are interchangeable. Our results provide a guideline on when employing open models to quantify closed queueing systems might be appropriate. That is, we shed some light on the following question: given that the number of repairmen is large how large the number of machines needs to be for the open approximation to be relevant on finite time intervals? In the context of process-level convergence, the Poisson and heavy-traffic limits are not interchangeable in the general case of single-server queues due to the difference in relevant time scales for the two limits. In particular, for single server queues, there

148

J. Pei, A. Motaei and P. Momčilović / Operations Research Letters 48 (2020) 147–151

exists a separation of time scales: service (repair) times occur on a faster time scale than the time scale associated with a single source (machine). On the other hand, in our model, repair and working times are associated with the same time scale; for a detailed discussion of relevant time scales see [14, Section 9.8]. Notation. Denote by D[0, ∞) the space of all real-valued functions on [0, ∞) that are right-continuous with left limits (r.c.l.l.), endowed with the standard J1 topology. The J1 metric is denoted by dJ1 (·, ·), |·| stands for the Euclidean metric on R, and the uniform metric u is defined by the supremum norm (for x ∈ D[0, ∞) and t ≥ 0)

∥x∥t = sup |x(s)|.

Fig. 1. A commutative diagram illustrating the interchange of Poisson (r → ∞) and QED (k → ∞) limits.

distributed according to the residual distribution of F r (G). Recall ¨ are defined by that the residual distributions F¨ r and G

0≤s≤t

The symbols ∗, ∧ and ∨ denote the convolution, minimum and maximum operators, respectively. Given a distribution function H, H (n) denotes the n-fold convolution of H with itself, i.e., H (2) = H ∗ H; H (0) ≡ 1. For x ∈ D[0, ∞) or x ∈ R, define x¯ = 1 − x, x+ = x ∨ 0 and x− = −(x ∧ 0) (that is, x = x+ − x− ). Let ⇒ denote weak convergence — for stochastic processes in D[0, ∞) and for P

random variables in R; → stands for convergence in probability. 2. Model Consider a sequence of machine repair models indexed by a pair (k, r); in such a model, the repair time distribution is G(·), the working time distribution is F r (·) = F (·/r), and the number of repairmen is k. The number of machines in the (k, r) model is denoted by nk,r . At any instant of time, a machine is either working, or it is broken. A machine works for a random amount of time distributed according to F r (with mean 1/λr := r /λ < ∞, F (0) = 0). Once broken, a machine requires a repair that lasts a random amount of time distributed according to G (with mean 1/µ < ∞, G(0) = 0). A machine enters a repair phase as soon as a repairman is available; repairmen service machines according to the FCFS rule. For convenience, define pr :=

λ λr = λr + µ λ + rµ

as the long-run fraction of time a machine would spend being broken (repaired) in the case it had a dedicated repairman. Working and repair times are independent both across time and machines/repairmen. In addition, the distributions G and F satisfy a technical condition (see e.g. [4,7]): lim sup

F (t) ∨ G(t) t

t ↓0

< ∞.

(1)

Let X k,r (t) be the number of broken machines (being repaired or awaiting service) in the (k, r) system at time t; X k,r := {X k,r (t), t ≥ 0}. Similarly, we let Xik,r (t) be an indicator of the ith machine not working in the (k, r) system at time t; then, we have k,r

X k,r =

n ∑

k,r

Xi ,

i=1 k,r

k,r

where Xi := {Xi (t), t ≥ 0}. Initially, at time t = 0, ξ k,r machines are broken in the (k, r) system, i.e., X k,r (0) = ξ k,r . In particular, ξ k,r ∧ k machines are being repaired (those with indices i = 1, . . . , ξ k,r ∧ k), (ξ k,r − k)+ machines are awaiting service (those with indices i = ξ k,r ∧ k + 1, . . . , ξ k,r ), and nk,r − ξ k,r machines are working (those with indices i = ξ k,r + 1, . . . , nk,r ). We assume that the remaining working (repair) times of machines working (being repaired) at time t = 0 are independent and

F¨ r (x) := λr

x

∫

F¯ r (s) ds 0

and

¨ := µ G(x)

x

∫

¯ ds. G(s)

(2)

0

We consider a sequence of machine repair models in the QED regime. Analogous to open systems in the QED regime, the capacity of the system is approximately equal to the offered load. In such a regime, the capacity and the offered load are related via the square-root rule: k − nk,r pr √ =: β k → β, nk,r pr p¯ r

(3)

as k → ∞, for all relevant r and some β ∈ (−∞, ∞); recall that p¯ r = 1 − pr . In (3), k and nk,r are natural numbers, and that fact defines a sequence of feasible r’s for a given k and βk ; observe that pr is a monotonically decreasing function in r. The scaled total number of broken machines is given by X k,r − k . Xˆ k,r := √ nk,r pr p¯ r We assume that the system is in the QED regime at time t = 0 (since we focus on diffusion rather than fluid limits). In particular, the sequence of random variables {ξ k,r } satisfies, as k → ∞,

ξ k,r − k √ ⇒ ξˆ ∞,r , nk,r pr p¯ r and ξˆ ∞,r ⇒ ξˆ , as r → ∞. In addition, one has ξ k,r ⇒ ξ k,∞ , as r → ∞, and

ξ k,∞ − k ⇒ ξˆ , √ k

(4)

as k → ∞. This ensures that the limit interchange holds for the sequence of initial conditions. Note that these limits and the assumption on the distribution of residual repair and working times at t = 0 (see (2)) specify the state of the system at time t = 0. The above assumptions ensure that the system remains in the QED regime (as k → ∞, for all r) on finite time intervals (the weak limit of Xˆ k,r is non-degenerate). 3. Limit interchange In this section, we discuss a relation between closed and open systems in the QED regime via a Poisson limit (the law of small numbers). Under such a limit, the number of machines (customers) is increasing, while the offered load and the service distribution are kept constant. We argue that the Poisson and QED limits are interchangeable, as illustrated in Fig. 1. k,r k,r Let Zi := {Zi (t), t ≥ 0} be an indicator process of whether machine i is broken in the case when a dedicated repairman is k,r k,r always available to it. Both Xi and Zi describe the behavior of the same machine i (a machine is described by two sequences of working and repair times). In particular, a machine i, ξ k,r ∧ k < i ≤ ξ k,r , is waiting repair at time t = 0 in the original model

J. Pei, A. Motaei and P. Momčilović / Operations Research Letters 48 (2020) 147–151 k,r

(Xi ), while its repair starts at t = 0 in the infinite-repairmen k,r k,r model (Zi ). Several functions can be associated with Zi . To this r end, let P1 (t), t ≥ 0, be the conditional probability of a machine being broken at time t, given that a repair process of the machine started at time t = 0. Then, P1r := {P1r (t), t ≥ 0} is given by

¯ ∗ Rr → G¯ , P1r = Rr − G ∗ Rr = G

(5)

r

as r → ∞, where R is the renewal function associated with the distribution H r := G ∗ F r , i.e., Rr = 1 + H r + H r ∗ H r + · · · (e.g., see [3, p. 286]). Similarly, let P¨1r (t) and P¨0r (t) be the conditional probabilities of a machine being broken at time t, given that it is being repaired at t = 0 (with the distribution ¨ and it is working at of the remaining repair time equal to G) t = 0 (with the distribution of the remaining working time equal to F¨ r ), respectively. Consequently, if P¨1r := {P¨1r (t), t ≥ 0} and P¨0r := {P¨0r (t), t ≥ 0}, one has

¨ ∗ F r ∗ Rr − G¨ ∗ Rr = 1 − G¨ ∗ F¯ r ∗ Rr → G¨¯ , P¨1r = 1 + G ¯ ∗ Rr → 0, P¨0r = F¨ r ∗ Rr − F¨ r ∗ G ∗ Rr = F¨ r ∗ G

(6) (7)

as r → ∞. We note that, due to the construction of stationary on–off processes in [6], pr P¨1r + p¯ r P¨0r = pr holds for all r. Moreover, if the distribution H r is nonlattice, than P1r (t) → pr , as t → ∞ [1, p. 173]; the same applies to P¨1r and P¨0r [9, p. 237]. Finally, based on the preceding and the indexing of the machines (see Section 2), it follows that

⎧ r k,r ⎨P¨1 (t), 1 ≤ i ≤ ξ ∧ k, k,r r k,r EZi (t) = P1 (t), ξ ∧ k < i ≤ ξ k,r , ⎩¨r P0 (t), ξ k,r < i ≤ nk,r . k,r

Note that EZi (t) depends on the machine index i, since i determines the state of a machine at time t = 0 in the original finite-repairmen system: machines 1 ≤ i ≤ ξ k,r ∧ k are being repaired, machines ξ k,r ∧ k < i ≤ ξ k,r are awaiting repair, and machines ξ k,r < i ≤ nk,r are working. Analogous to X k,r and Xˆ k,r , define k,r

Z

k,r

:= {Z

k,r

(t), t ≥ 0} :=

n ∑

k,r Zi

and

Zˆ

k,r

i=1

Z k,r − nk,r pr

:= √

nk,r pr p¯ r

.

Zˆ k,r ⇒ Zˆ ∞,r ≡ Yˆ ∞,r + (ξˆ ∞,r + β − Yˆ ∞,r (0))(P¨1r − P¨0r ) (8)

where ξˆ ∞,r and Yˆ ∞,r := {Yˆ ∞,r (t), t ≥ 0} are independent, and Yˆ ∞,r is a stationary centered Gaussian process with the covariance function

EYˆ ∞,r (t)Yˆ ∞,r (t + s) =

P¨1r (s) − pr 1 − pr

,

t , s ≥ 0.

(9)

The last process we define is Zˆ = {Zˆ (t), t ≥ 0}, defined by

( ) ( ) ¨¯ + ξˆ + G¯ − G¨¯ , Zˆ := Yˆ + ξˆ + β − Yˆ (0) G

Definition 1. Let P be either G or P¯ 1r = 1 − P1r . The mapping ϕP : D[0, ∞) → D[0, ∞) is such that ϕP (x), for each x ∈ D[0, ∞), is the unique solution y to t

∫

y+ (t − s) dP(s).

y(t) = x(t) + 0

Two limits can be obtained for a two-dimensional sequence

{Xˆ k,r } (see Fig. 1). First, [7] implies, as k → ∞, Xˆ k,r ⇒ Xˆ ∞,r ≡ ϕP¯ r (Zˆ ∞,r − β ),

(11)

1

where Zˆ ∞,r is defined in (8). That is, Xˆ ∞,r is a QED limit of a sequence of many-server systems. Furthermore, one can let r → ∞ in that limit to complete the upper-right part of the diagram in Fig. 1. The following limit justifies Xˆ ∞,r ⇒ Xˆ , as r → ∞. Lemma 1.

We have ϕP¯ r (Zˆ ∞,r − β ) ⇒ ϕG (Zˆ − β ), as r → ∞. 1

Proof. In the first step, we argue ∥P¯ 1r − G∥t → 0, as l → ∞. To this end, (5) implies P¯ 1r = G − P1r ∗ G ∗ F r , and thus

∥P¯ 1r − G∥t = ∥P1r ∗ G ∗ F r ∥t ≤ G ∗ F r (t) ≤ F r (t), where the second inequality is due to P1r (t) ∈ [0, 1], for all t, and monotonicity of distribution functions. The desired result follows from F r (t) = F (t /r) → 0, as r → ∞ (recall that F (0) = 0). Here, we also note that (5) and (7) render P¨0r = F¨ r ∗ P1r , which results in ∥P¨0r ∥t ≤ F¨ r (t) ≤ λt /r → 0, as r → ∞ (see (2)). Similarly, (6), monotonicity of distribution functions and Rr ∗ G ∗ F r = Rr − 1 yield

∥P¨1r − G¨¯ ∥t = ∥G¨ − G¨ ∗ F¯ r ∗ Rr ∥t ≤ ∥F r − F¯ r ∗ (Rr − 1)∥t ≤ F r (t) + Rr (t) − 1 ≤ F r (t) + 1/F¯ r (t) − 1 → 0, as r → ∞, where Rr (t) ≤ 1/F¯ r (t) was used to obtain the last inequality. Observe that this implies, as r → ∞,

   ¯ P¨1r − pr  G¨ −  → 0;  1 − pr t

Under (1), a centered and scaled version of the infinite-repairmen process satisfies [7], as k → ∞:

+ (ξˆ ∞,r )+ (P1r − P¨1r ),

149

that is, the covariance functions of Yˆ and Yˆ ∞,r converge. The second step of the proof is to show Zˆ ∞,r ⇒ Zˆ , as r → ∞. From the Gaussian nature of Yˆ ∞,r and Yˆ , the first step of the proof and independence of the following components, it follows that

¨¯ , G¯ − G¨¯ , ξˆ ), (Yˆ ∞,r , P¨1r − P¨0r , P1r − P¨1r , ξˆ ∞,r ) ⇒ (Yˆ , G as r → ∞, in (D3 [0, ∞) × R, u3 × |·|). Given that a (D3 [0, ∞) × R, u3 × |·|) → (D3 [0, ∞), u3 ) function (x1 , x2 , x3 , x4 ) ↦ → (x1 , (x4 + β − x1 (0))x2 , x+ 4 x3 ) is continuous, the continuous mapping theorem, and (10) yield Zˆ ∞,r ⇒ Zˆ , as r → ∞. Now, we are ready for the last step of the proof. Results in [8] and the previous step of this proof imply that, as r → ∞,

ϕG (Zˆ ∞,r − β ) ⇒ ϕG (Zˆ − β ). (10)

where Yˆ := {Yˆ (t), t ≥ 0} is a stationary centered Gaussian ¨ process with EYˆ (t)Yˆ (t + s) = 1 − G(s), t , s ≥ 0. (Yˆ is a stationary limiting (centered and scaled) M/G/∞ process defined by the service distribution G.) The number-in-system process can be expressed in terms of the corresponding infinite-server process via the following nonlinear convolution operator defined below (the operator is well-defined due to [8] and [7]).

(12)

In view of (12) (see also Theorem 11.4.5 in [14, p. 379]), in order to complete the proof, it is sufficient to show that, as r → ∞,

∥ϕP¯ r (Zˆ ∞,r − β ) − ϕG (Zˆ ∞,r − β )∥t → 0. P

(13)

1

For notational simplicity, let y˜ r = ϕP¯ r (Zˆ ∞,r − β ) and yr = 1

ϕG (Zˆ ∞,r −β ). Suppose first that G is a non-degenerate distribution, so that there exists a δ > 0 and an ϵ < 1 as in [7, Lemma 7]. Then, one has

∥˜yr − yr ∥δ ≤ ∥P¯ 1r − G∥δ ∥˜yr ∥δ + ϵ∥˜yr − yr ∥δ ,

150

J. Pei, A. Motaei and P. Momčilović / Operations Research Letters 48 (2020) 147–151

which, after letting r → ∞ and invoking the first step of the P

proof, implies ∥˜yr − yr ∥δ → 0. Repeating this argument multiple times yields (13) in the case of non-degenerate G. On the other hand, if G is concentrated on cG , then one has ∥˜yr − yr ∥t = 0 (almost surely, for all r), for 0 ≤ t < cG , since P¯ 1r (t) = G(t) for those t. Next, for cG ≤ t < 2cG , it follows that

∥˜yr − yr ∥t ≤ ∥P¯ 1r − G∥t ∥˜yr ∥t + ∥˜yr − yr ∥t −cG ,

1 ≤ i ≤ ξ k,r and l = 1 – {˜ai,1 , 1 ≤ i ≤ ξ k,r } are i.i.d., distributed according to F¨ r , and independent of all other variables. Since the arrival process to the newly constructed system is a sum of r independent renewal processes, the arrival rate to the system is given by

λr nk,r = (λr + µ)pr nk,r → µθ k ,

and therefore ∥˜yr − yr ∥t → 0, as r → ∞, for 0 ≤ t < 2cG . Iterating this procedure multiple times yields (13) in the case of degenerate G. Thus, (13) holds. □ P

Next, we consider the bottom-left part of Fig. 1. Let X k,∞ be the number-in-system process for an open system with k servers, the service distribution G√and Poisson arrivals with rate µθ k , where θ k solves k = θ k +β k θ k ; the number of customers in the system at t = 0 is ξ k,∞ ; residual service times of ξ k,∞ ∧ k customers in ¨ service are i.i.d. with the distribution G.

√

X

ξ k,r ∧k

(t) =

∑

r ,k

k,r Xi (t)

+

k,r Xi (t)

i=ξ k,r ∧k+1

i=1

+

n ∑

k,r

k,r

∞ ∑

i=ξ k,r +1

1{0≤t −∑j−1 (a l =0

j=1

i,l+1 +si,l +wi,l )
,

= 1{0≤t
∞ ∑

1{0≤t −∑j

,

and, for machines ξ k,r < i ≤ nk,r , k,r

Xi (t) =

∞ ∑

1{0≤t −a

∑j−1

i,j −

j=1

l =1

(ai,l +sj,l +wj,l )
;

in the last equation, ai,1 is distributed according to F¨ r . Next, we construct an alternative queueing system coupled with the original one: let X˜ k,r = {X˜ k,r (t), t ≥ 0} be an open queueing system with i.i.d. service times distributed according to G, while arrivals occur according to a sum of r stationary renewal processes defined by distribution F r ; the content of the newly created system at time t = 0 is identical to the content of the original system at t = 0. Formally, for t ≥ 0, X˜ k,r (t) =

ξ k,r ∧k

∑

k,r

ξ ∑

1{0≤t
1{0≤t
i=ξ k,r ∧k+1

i=1 k,r

+

n ∞ ∑ ∑

1{0≤t −∑j

i=1 j=1

=: X k,r (t) − ∆k,r (t),

( k 1−

(a +si,l +wi,l )
j=1

(

Given that the initial number of customers in the system at time t = 0 obeys (4), and the utilization θ k /k in the limiting (as r → ∞) system with k servers (M/G/k) satisfies (see (3))

√

where wi,l is the waiting time before repair time si,l , which is followed by working time ai,l+1 ; here, wi,0 ≡ 0 and si,0 is distributed ¨ Similarly, for machines ξ k,r ∧ k < i ≤ ξ k,r , according to G. k,r Xi (t)

i=1

)nk,r

as r → ∞, where the limit follows from (1), (15), dominated convergence theorem and ξ k,r ⇒ ξ k,∞ , as r → ∞. Hence, ∆k,r ⇒ 0, as r → ∞, and the statement of the lemma follows. □

Xi (t);

the three sums correspond to the machines being repaired, awaiting repair and working at time t = 0, respectively. In particular, for machines 1 ≤ i ≤ ξ k,r ∧ k, Xi (t) = 1{0≤t
k

)ξ k,r ≥ 1 − F¨ r (T ) F r (T ) E 1 − F r (T ) ( )nk,r k,r = 1 − F¨ (T /r) F (T /r) E (1 − F (T /r))ξ → 0,

r ,k

ξ ∑

k

i=1

(

Proof. Recall the construction of the process X k,r from Section 2. The number of broken machines at time t ≥ 0 can be expressed as k,r

k

⎡ k,r ⎤ ⎡ k,r ⎤ ξ n ⋀ ⋀ [ k,r ] P ∥∆ ∥T = 0 ≥ P ⎣ (a˜ i,1 + a˜ i,2 ) > T ⎦ P ⎣ ai,1 > T ⎦

We have Xˆ k,r ⇒ Xˆ k,∞ ≡ (X k,∞ − k)/ k, as r → ∞.

Lemma 2.

(15)

√

as r → ∞, where θ solves k = θ + β θ due to (3). Moreover, for a fixed k, the sum of (scaled) renewal processes (the arrival process to X˜ k,r ) converges weakly, as r → ∞, to a Poisson process with rate µθ k [2, Section 3]. This fact, coupled with the queueing operator (that maps arrival and service times to the occupancy process) being continuous [12, Section 4], yields X˜ k,r ⇒ X k,∞ , as r → ∞, where X k,∞ is the M/G/k process described earlier. In view of (14), it remains to show ∆k,r := {∆k,r (t), t ≥ 0} ⇒ 0, as r → ∞. To this end, k

a˜
(14)

where w ˜ i,j is the waiting time of the customer with arrival time ∑j ˜ ˜ i,l = ai,l , except for a i , l and service requirement si,j ; here, a l=1

µθ k µk

)

√ = βk

θk k

→ β,

as k → ∞, the sequence of open systems indexed by k is in the QED regime. The main result in [8] implies, as k → ∞, Xˆ k,∞ − k

√

k

⇒ Xˆ ≡ ϕG (Zˆ − β ),

(16)

where Zˆ is as in (10). The result in [8] is stated in terms of a timechanged Brownian bridge and a non-stationary limiting (centered and scaled) infinite-server process; those two terms correspond to customers initially (at time t = 0) present in the system and those arriving after t = 0, respectively. However, considering all customers together results in (16) and (10). We note that comparison Xˆ and Xˆ ∞,r via (16) and (11) is not straightforward, in general. The reason is that the difference in Xˆ and Xˆ ∞,r is not only due to the different nonlinear operators ϕG and ϕP¯ r , but 1

different correlation structures of Zˆ and Zˆ ∞,r (or rather different covariance functions of Yˆ and Yˆ ∞,r ) as well. To conclude, the diagram in Fig. 1 provides a justification for approximating machine repair model in the QED regime with its open analogue in the same heavy-traffic regime. Performance of a machine repairmen model with many repairmen in a finite time interval [0, t ] approaches performance of the corresponding open model (M/G/k), as the number of machines increases (while the load is kept constant). Informally, our result indicates that such an approximation might be appropriate when the following quantity are ‘‘small": (i) ∥P¯ 1 − G∥t (compare (11) and (16)); (ii) ∥P¨1 − P¨0 − 1 + G¨ ∥t and ∥P1 − P¨1 − G¨ − G∥t (compare (8) and (10)); ¨ ∥t (compare the covariance functions (iii) ∥(P¨1 − p)/¯p − 1 + G for Yˆ ∞,r and Yˆ ). Assuming that the mean of working times is 1/λ, these conditions are satisfied when λ/µ ≪ 1, λt ≪ 1 and H(t) ≪ 1.

J. Pei, A. Motaei and P. Momčilović / Operations Research Letters 48 (2020) 147–151

Acknowledgment The work of P.M. was supported in part by the NSF, USA Award CMMI-1362630. References [1] S. Asmussen, Applied Probability and Queues, second ed., Springer-Verlag, New York, NY, 2003. [2] E. Çinlar, Superposition of point processes, in: P.A.W. Lewis (Ed.), Stochastic Point Processes: Statistical Analysis, Theory and Application, Wiley, New York, NY, 1972, pp. 594–606. [3] E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, NJ, 1975. [4] D. Gamarnik, D. Goldberg, Steady-state GI/GI/N queue in the Halfin-Whitt regime, Ann. Appl. Probab. 23 (6) (2013) 2382–2419. [5] L. Green, S. Savin, M. Murray, Providing timely access to care: what is the right patient panel size? Jt. Comm. J. Qual. Patient Saf. 33 (4) (2007) 211–218.

151

[6] D. Heath, S. Resnick, G. Samorodnitsky, Heavy tails and long range dependence in on/off processes and associated fluid models, Math. Oper. Res. 23 (1) (1998) 145–165. [7] P. Momčilović, A. Motaei, An analysis of a large-scale machine repair model, Stoch. Syst. 8 (2) (2018) 91–125. [8] J. Reed, The G/GI/N queue in the Halfin-Whitt regime, Ann. Appl. Probab. 19 (6) (2009) 2211–2269. [9] S. Resnick, Adventures in Stochastic Processes, Birkhäuser, Boston, MA, 1992. [10] F. de Véricourt, O. Jennings, Dimensioning large-scale membership services, Oper. Res. 56 (1) (2008) 173–187. [11] F. de Véricourt, O. Jennings, Nurse staffing in medical units: a queueing perspective, Oper. Res. 59 (6) (2011) 1320–1331. [12] W. Whitt, The continuity of queues, Adv. Appl. Probab. 6 (1) (1974) 175–183. [13] W. Whitt, Open and closed models for networks of queues, AT & T Bell Lab. Tech. J. 63 (9) (1984) 1911–1979. [14] W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, New York, NY, 2002.