On a doubly dynamically controlled supermarket ... - Semantic Scholar

Computers & Operations Research 55 (2015) 76–87

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On a doubly dynamically controlled supermarket model with impatient customers Quan-Lin Li n, Ye Du, Guirong Dai, Meng Wang School of Economics and Management Sciences, Yanshan University, Qinhuangdao 066004, PR China

art ic l e i nf o

a b s t r a c t

Available online 16 October 2014

In this paper, we provide a key generalization of the supermarket model both from the impatient customers and from a doubly dynamic control, which may also be related to the size-based scheduling through the centered management of the customer resource as well as the total service ability. We first use an infinite-dimensional Markov process to express the states of this supermarket model, and set up an infinite-dimensional system of differential equations satisfied by the expected fraction vector. Then we use the operator semigroup to provide a mean-field limit for the sequence of infinite-dimensional Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Finally, we provide an effective and efficient algorithm for computing the fixed point of the infinite-dimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model. Also, some numerical examples are given to demonstrate how the performance measures depend on some crucial parameters of this supermarket model. We believe that the mean-field method developed in this paper will be useful and effective for analyzing more complicated supermarket models in many practical areas. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Supermarket model Impatient customer Operator semigroup Mean-field limit Fixed point Performance analysis

1. Introduction Dynamic randomized load balancing is often referred to as the supermarket model. The supermarket models have a wide range of practically applicable areas such as call centers, health care, computer networks, production/inventory systems, and transportation networks. Recently, the supermarket models have been analyzed by means of queueing methods as well as Markov processes. For a simple supermarket model, Vvedenskaya et al. [50] applied the operator semigroup and mean-field limit to compute the stationary queue length distribution of any queueing process, and obtained its doubly exponential decay tail which is a substantial improvement of system performance over that in the ordinary M/M/1 queue. At nearly the same time, Mitzenmacher [37] also analyzed the same supermarket model in terms of the density-dependent jump Markov processes. Turner [48] provided a martingale approach to further discuss the supermarket model. The path space evolution of the supermarket model was studied by Graham [23,24] who showed that starting from independent initial states, as N-1 the queues of the limiting process evolve

n

Corresponding author. Tel.: þ 86 335 8532035. E-mail address: [email protected] (Q.-L. Li).

http://dx.doi.org/10.1016/j.cor.2014.10.004 0305-0548/& 2014 Elsevier Ltd. All rights reserved.

independently. Luczak and McDiarmid [33] showed that the length of the longest queue scales as ðlog log NÞ=log d þ Oð1Þ. Certain generalization of the supermarket model has been explored in studying various variations, for example, modeling more crucial factors by Vvedenskaya and Suhov [51], Mitzenmacher [38], Mitzenmacher et al. [39], Bramson et al. [10–12], Li and Lui [30,31], Li et al. [32,29] and Li [27,28]; fast Jackson networks by Martin and Suhov [35], Martin [34] and Suhov and Vvedenskaya [46]. Queues with impatient customers represent a wide range of service systems in which customers may become impatient when they do not receive service fast enough, and they are always useful in modeling many practical situations such as banks, hospitals, supermarkets, supply chains and transportation systems. Readers may refer to some important publications for more details, among which are the M/M/1 queue with impatient customers by Choi et al. [16], Brandt and Brandt [14] and Altman and Yechiali [1]; the M/M/c queue with impatient customers by Stanford [44], Bhattacharya and Ephremides [5], Boxma and de Waal [7], Movaghar [40], Brandt and Brandt [13], Boots and Tijms [6], Yechiali [52] and Perel and Yechiali [43]; the GI/M/s queue with impatient customers by Teghem [47], Swensen [45] and Perry et al. [42]; the M/G/ 1 queue with impatient customers by de Kok and Tijms [21], Bae et al. [4], Perry and Asmussen [41], Martin and Artalejo [36], Boxma et al. [8], Iravani and Balcıoğlu [25] and Boxma et al. [9];

Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

and the GI/G/1 queue with impatient customers by Daley [19], Baccelli and Hébuterne [3] and Baccelli et al. [2]. The matrixanalytic method is applied to study queues with impatient customers, e.g., see Combé [18], Choi et al. [17], Van Velthoven et al. [49] and Chakravarthy [15]. In applications of queues with impatient customers, examples include Kaspi and Perry [26] for inventory systems, Zohar et al. [54] for call centers, and Zeltyn and Mandelbaum [53] for telecommunication networks. The purpose of this paper is to provide a key generalization of the supermarket model both from the impatient customers and from a doubly dynamic control. The generalized supermarket model is described as a queueing network which consists of one server and N waiting lines with impatient customers under two centered management modes: Each arriving customer joins the shortest one among the d1 randomly selected waiting lines (JSQ (d1)), and the server provides its service in the longest one among the d2 randomly selected waiting lines (SLQ (d2)). Firstly, such two centered management modes may be related to the size-based scheduling both for a suitable allocation of customer resource and for a total use of multiple service units. Then our numerical examples indicate that for improving system performance, the SLQ(d2) controlling the service processes is more effective than the JSQ(d1) controlling the arrival processes. Finally, it is necessary to analyze the useful relations among some special cases. When d2 ¼ 1 and the impatience rate θ ¼0, this supermarket model is equivalently degraded to an ordinary supermarket model, that is, each waiting line has a server with exponential service times of rate μ. When d1 ¼ 1 and d2 ¼ 1, this supermarket model is equivalent to a system of N independent M/M/1 queues with impatient customers. The main contributions of this paper are twofold. The first one is to provide a key generalization of the supermarket model both from the impatient customers and from a doubly dynamic control. We use the operator semigroup to provide the mean-field limit for the sequence of infinite-dimensional Markov processes, and give a computational framework for organizing a Lipschitzian condition, which is always a key for proving existence and uniqueness of solution to the infinite-dimensional system of differential equations by means of the Picard approximation. The second contribution of this paper is to provide an effective and efficient algorithm for computing the fixed point, which leads to performance analysis of this supermarket model. Also, some numerical examples are used to indicate how the performance measures depend on some crucial parameters of this supermarket model. It is worthwhile to note that the results given in this paper are lucky from analysis of a key generalized supermarket model, and also improve some numerical research from the double choice numbers d1 Z 1 and d2 Z 1, since only one choice of the parameter, i.e., d ¼2 was always assumed in the literature, e.g., see Vvedenskaya and Suhov [51] and Mitzenmacher et al. [39]. Note that the double choice numbers d1 Z 1 and d2 Z 1 are respectively related to the size-based scheduling through the centered management of the customer resource as well as the total service ability, thus this paper provides some new highlight on analyzing more complicated supermarket models from the size-based scheduling, e.g., see Dell'Amico et al. [20]. The remainder of this paper is organized as follows. In Section 2, we describe a supermarket model with impatient customers under a doubly dynamic randomized load balancing control, and use an infinite-dimensional Markov process to express the state of the supermarket model. In Section 3, we set up an infinite-dimensional system of differential equations, which is satisfied by the expected fraction vector. In Section 4, we apply the operator semigroup to provide a mean-field limit for the sequence of infinite-dimensional Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-

77

dimensional system of limiting differential equations. In Section 5, we provide a computational framework for organizing a Lipschitzian condition, which is always a key for proving existence and uniqueness of solution to the infinite-dimensional system of differential equations by means of the Picard approximation. In Section 6, we provide an effective and efficient algorithm for computing the fixed point. In Section 7, we use the fixed point give performance analysis of this supermarket model, and provide some numerical examples to analyze how the performance measures depend on some crucial parameters of this supermarket model. Some concluding remarks are given in the final section.

2. A supermarket model with impatient customers In this section, we describe a supermarket model with impatient customers, which consists of one server and N waiting lines under a doubly dynamic randomized load balancing control. We use the fraction vector of waiting lines with at least k customers for k Z 0 to organize an infinite-dimensional Markov process who expresses the state of the supermarket model at time t Z 0. 2.1. Model description Let us describe the supermarket model with impatient customers, which consists of one server and N waiting lines under a doubly dynamic randomized load balancing control, where the random factors of this supermarket model are listed as follows: The arrival process and its dynamically control: Customers arrive at the supermarket model as a Poisson process with arrival rate N λ for λ 4 0. Upon arrival, each customer chooses d1 Z 1 waiting lines from the N waiting lines independently and uniformly at random, and joins the one whose queue length is the shortest among the d1 selected waiting lines. If there is a tie, waiting lines with the shortest queue are chosen randomly. The service process and its dynamical control: There is one server working among the N waiting lines in this system, and the service times are i.i.d. and is exponential with service rate N for μ 4 0. The server always chooses d2 Z 1 waiting lines independently and uniformly at random from the N waiting lines, and provides its service in the one whose queue length is the longest among the d2 selected waiting lines. The impatient process: The waiting customers excluding the one in service are impatient, and each of them has an exponential sojourn time with impatience rate N θ for θ 4 0. Obviously, if there is a customer at the server and k  1 customers in the waiting room (note that there are k customers in the system), then ðk  1ÞN θ is the total impatience rate due to the independently impatient behavior of the k  1 customers. We assume that all the random variables defined above are independent of each other. Fig. 1 provides a physical illustration of the supermarket model with impatient customers.

Waiting Line 1 1 Arrivals

1

2

Waiting Line 2

d1 1

Waiting Line 3

2

N

Each customer probes d 1 servers

3 d2 1

d1

Waiting Line N

One server N

d2

Fig. 1. A physical illustration of the supermarket model with impatient customers.

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Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

In this supermarket model depicted in Fig. 1, the choice number d1 Z1 expresses the route scheduling of arriving customers and the choice number d2 Z 1 shows the effective use of total service ability. When d2 ¼ 1, this supermarket model is equivalently degraded to an ordinary supermarket model, where each waiting line has a server with exponential service times of rate μ. It is easy to see from the above doubly dynamically control for the arrival and service processes that the generalized supermarket model may be related to the size-based scheduling through the centered management of the customer resource as well as the total service ability. The following lemma provides a sufficient condition under which the supermarket model with impatient customers is stable.

Since for a fixed pair ðt; NÞ, the sample value of UðNÞ ðtÞ is in the set

 0 1 N1 N 1 N1 ; ; …; ; ¼ 0; ; …; ;1 ; N N N N N N there exists a larger positive integers K such that ðNÞ ðNÞ U 0ðNÞ ðtÞ ZU ðNÞ 1 ðtÞ Z ⋯ Z U K ðtÞ 40 and U k ðtÞ ¼ 0 for k ZK þ 1. To study the infinite-dimensional Markov process fUðNÞ ðtÞ : t Z 0g, we write the expected fractions as follows: uðNÞ ðtÞ ¼ E½U ðNÞ ðtÞ k k and ðNÞ ðNÞ ðNÞ uðNÞ ðtÞ ¼ ðuðNÞ 0 ðtÞ; u1 ðtÞ; u2 ðtÞ; u3 ðtÞ; …Þ:

It is clear that Lemma 1. For the supermarket model with impatient customers which consists of one server and N waiting lines under a doubly dynamic randomized load balancing control, it is stable if ρ ¼ λ=μ o 1. Proof. If d1 ¼ d2 ¼ 1, then this supermarket model is equivalent to a system of N independent M/M/1 queues with impatient customers. It is well-known that the M/M/1 queue with impatient customers is stable if ρ o1. Using a coupling method, as given in Martin and Suhov [35, Theorems 4 and 5], this supermarket model with impatient customers is stable if ρ o 1. This completes the proof. □ Remark 1. In a practical supermarket model, the impatient behavior of customers may have different classes, such as, Class one: The impatient behavior only exists for customers in the waiting room. Once a customer enters a server, it must finish its complete service and then leave this system. Such an impatient example can be seen in, such as, bank, supermarket, and hospital. Class two: the impatient behavior exists for customers with a time-length constraint, thus the impatient customers may leave this system both from the waiting room and from the server. Such an impatient example can be seen in either meeting, interview or perishable product. 2.2. An infinite-dimensional Markov process For k Z0, we denote by nk ðtÞ the numbers of waiting lines with at least k customers at time t Z 0. Clearly, n0 ðtÞ ¼ N and 0 r nk ðtÞ r N for k Z 1. We write that for k Z 0 ðtÞ ¼ U ðNÞ k

nk ðtÞ ; N

which is the fraction of waiting lines with at least k customers at time t Z 0. Let ðNÞ UðNÞ ðtÞ ¼ ðU 0ðNÞ ðtÞ; U ðNÞ 1 ðtÞ; U 2 ðtÞ; …Þ:

Then the state of this supermarket model with impatient customers is described as an infinite-dimensional stochastic process fUðNÞ ðtÞ : t Z 0g. Since the arrival process is Poisson and the service and impatient times are exponential, fUðNÞ ðtÞ : t Z 0g is an infinitedimensional Markov process whose state space is given by EN ¼ fuðNÞ : 1 ¼ NuðNÞ k

ðNÞ ðNÞ ðNÞ uðNÞ 0 Z u1 Zu2 Zu3 Z ⋯ Z 0;

is a nonnegative integer for k Z 0g:

For a fixed pair ðt; NÞ with t Z 0 and N ¼ 1; 2; 3; …, it is easy to see from the stochastic order that U ðNÞ ðtÞ Z U ðNÞ ðtÞ for k Z 0. This gives k kþ1 that ðNÞ ðNÞ ðNÞ 1 ¼ U ðNÞ 0 ðtÞ Z U 1 ðtÞ Z U 2 ðtÞ Z U 3 ðtÞ Z⋯ Z 0:

ðNÞ 1 ¼ u0ðNÞ ðtÞ Z uðNÞ 1 ðtÞ Z u2 ðtÞ Z⋯:

3. The system of differential equations In this section, for this supermarket model with impatient customers, we set up an infinite-dimensional system of differential equations satisfied by the expected fraction vector uðNÞ ðtÞ for t Z 0 by means of the technique of tail probabilities, e.g., see Li [28]. In the supermarket model with impatient customers, to set up an infinite-dimensional system of differential equations satisfied by the expected fraction vector uðNÞ ðtÞ, we need to determine the expected change in the number of waiting lines with at least k customers over a small time period ½0; dtÞ. To that end, our computation contains the following three parts: (a) The arrival process: The rate that any arriving customer selects d1 waiting lines from the N waiting lines independently and uniformly at random, joins the one whose queue length is the shortest, and the queue lengths of the other selected d  1 waiting line are not shorter than k  1 is given by d1

ðNÞ ðNÞ m ðNÞ d1  m Nλ ∑ C m dt ¼ N λf½uðNÞ ðtÞd1  ½uðNÞ ðtÞd1 g dt; d1 ½uk  1 ðtÞ  uk ðtÞ ½uk ðtÞ k1 k m¼1

ð1Þ Cm d

where ¼ d!=½m!ðd mÞ! is a binomial coefficient. Note that in the d1 selected waiting lines and 1 r m rd1 , ½ukðNÞ ðtÞ  uðNÞ ðtÞm is 1 k the probability that any arriving customer who can only choose one server makes m independent selections from the m servers with the shortest queue length k  1, while ½uðNÞ ðtÞd1  m is the k probability that there are d1  m selected waiting lines whose queue length are not shorter than k. (b) The service process: The server always chooses d2 Z1 waiting lines independently and uniformly at random from the N waiting lines, and provides its service in the one whose queue length is the longest among the d2 selected waiting lines. Based on this, the rate that the server goes to one waiting line with the longest queue length k among the d2 selected waiting lines, and the queue lengths of the other selected d2  1 waiting lines are not longer than k, is given by d2

ðNÞ ðNÞ ðNÞ m d2  m dt Nμ ∑ C m d2 ½uk ðtÞ  uk þ 1 ðtÞ ½1  uk ðtÞ m¼1

ðtÞd2  ½1 uðNÞ ðtÞd2 g dt: ¼ N μf½1  uðNÞ kþ1 k

ð2Þ

Note that in the d2 selected waiting lines and 1 r m r d2 , ½ukðNÞ ðtÞ  uðNÞ ðtÞm is the probability that the server makes m kþ1 independent selections from the m waiting lines with the longest queue length k, while ½1  uðNÞ ðtÞd2  m is the probability that there k are d2  m selected waiting lines whose queue length are not longer than k  1. (c) The impatient process: The rate that one impatient customer cannot wait for his service and leave this system from its waiting

Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

For k Z 0, we write

line queued by k customers is given by Nðk  1Þθ

½ukðNÞ ðtÞ uðNÞ ðtÞ kþ1

ð3Þ

dt:

ðtÞd1  ½uðNÞ ðtÞd1 g dt dE½nkðNÞ ðtÞ ¼ N λf½uðNÞ k1 k

μ

d p ðtÞ ¼  λp0 ðtÞ þ μp1 ðtÞ; dt 0

dt

θ

ð10Þ

for k Z 2 d p ðtÞ ¼  ½λ þ μ þ ðk  1Þθpk ðtÞ þ λpk  1 ðtÞ þ ðμ þ kθÞpk þ 1 ðtÞ: dt k

ðtÞ  uðNÞ ðtÞ;  ðk  1Þθ½uðNÞ k kþ1

ð4Þ

ðtÞ ¼ E½nðNÞ ðtÞ=N. which is due to uðNÞ k k Using some similar analysis to Eqs. (4), we can obtain an infinite-dimensional system of differential equations satisfied by ðNÞ the expected fraction vector uðNÞ ðtÞ ¼ ðu0ðNÞ ðtÞ, uðNÞ 1 ðtÞ; u2 ðtÞ; …Þ as follows: n o n o d ðNÞ u1 ðtÞ ¼ λ ½uðNÞ ðtÞd1  ½uðNÞ ðtÞd1  μ ½1  uðNÞ ðtÞd2  ½1  u1ðNÞ ðtÞd2 0 1 2 dt ð5Þ and for k Z 2 n o n d ðNÞ uk ðtÞ ¼ λ ½uðNÞ ðtÞd1  ½uðNÞ ðtÞd1  μ ½1  ukðNÞ ðtÞd2 k  1 k þ1 dt ðtÞd2 g  ðk  1Þθ½uðNÞ ðtÞ  uðNÞ ðtÞ;  ½1  uðNÞ k k kþ1

t Z 0;

ð11Þ

Let 1

Q k ðtÞ ¼ ∑ pi ðtÞ; i¼k

k Z 0;

Then it follows from (9) to (11) that d Q ðtÞ ¼ λ½Q 0 ðtÞ  Q 1 ðtÞ  μ½Q 1 ðtÞ  Q 2 ðtÞ; dt 1

ð12Þ

for k Z 2 d Q ðtÞ ¼ λ½Q k  1 ðtÞ  Q k ðtÞ  ½μ þðk  1Þθ½Q k ðtÞ  Q k þ 1 ðtÞ; dt k

ð13Þ

with the boundary condition Q 0 ðtÞ ¼ 1: ð6Þ

with the boundary condition

ð14Þ

It is easy to see that Eqs. (12) and (13) are the same as Eqs. (5) and (6) for d1 ¼ d2 ¼ 1.

ð7Þ 4. A mean-field limit

and the initial conditions uðNÞ ð0Þ ¼ g k ; k

ð9Þ

d p ðtÞ ¼  ðλ þ μÞp1 ðtÞ þ λp0 ðtÞ þ ðμ þ θÞp2 ðtÞ; dt 1

This gives that for k Z 1 n o d ðNÞ u ðtÞ ¼ λ ½uðNÞ ðtÞd1  ½uðNÞ ðtÞd1 k1 k dt k ðtÞd2  ½1  ukðNÞ ðtÞd2 g  μf½1  ukðNÞ þ1

uðNÞ 0 ðtÞ ¼ 1;

pk ðtÞ ¼ PfNðtÞ ¼ kg: Using Fig. 2, we obtain

Based on the above analysis from (1) to (3), we obtain  N f½1  uðNÞ ðtÞd2 ½1  uðNÞ ðtÞd2 g kþ1 k ðtÞ  ukðNÞ ðtÞ dt:  Nðk  1Þ ½uðNÞ k þ1

79

k Z 0;

ð8Þ

where 1 ¼ g 0 Z g 1 Z g 2 Z ⋯ Z 0: In the remainder of this section, we consider a special supermarket model with d1 ¼ d2 ¼ 1, that is, the M/M/1 queue with impatient customers. We denote by NðtÞ the number of customers in the M/M/1 queue with impatient customers. Then NðtÞ ¼ 0; 1; 2; …. It is easy to see that fNðtÞ : t Z 0g is a Markov chain whose state transition relation is depicted in Fig. 2.

Idle 0

Working 1

In this section, we use the operator semigroup to provide a mean-field limit for the sequence fUðNÞ ðtÞ : t Z 0g of infinitedimensional Markov processes, and show that this sequence of Markov processes can asymptotically approach a single trajectory identified by the unique and global solution to the infinitedimensional system of limiting differential equations. ðNÞ ðNÞ ðNÞ For the vector uðNÞ ¼ ðuðNÞ 0 ; u1 ; u2 ; u3 ; …Þ, we write ~ ¼ fuðNÞ : 1 ¼ uðNÞ Z uðNÞ Z uðNÞ Z ⋯ Z 0; Ω N 0 1 2 NukðNÞ is a nonnegative integer for k Z0g:

Working 2

Working 3

3

2

Service process

Arrival process

One customer in the server

k 1

Impatient process

There are k customers in the system

(k-1) customers in the waiting room

Fig. 2. State transition relation in the M/M/1 queue with impatient customers.

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Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

and

ΩN ¼ fu

ðNÞ

:u

ðNÞ

~ AΩ

N

and u

ðNÞ

e o þ 1g;

where e is a column vector of ones with a suitable dimension in the context. For the vector u ¼ ðu0 ; u1 ; u2 ; u3 ; …Þ, we set ~ ¼ fu : 1 ¼ u Zu Z u Z ⋯ Z 0g Ω 0 1 2 and ~ and ueo þ 1g: Ω ¼ fu : u A Ω ~ and Ω ⫋Ω ~ ⫋Ω ~. Obviously, ΩN ⫋Ω⫋Ω N N ~ , we take a metric In the vector space Ω
 ju  u0 j ~: ρððu; u0 Þ ¼ sup k k ; u; u0 A Ω kþ1 kZ0

and h  i e  ∂ f ðgÞ; N f g  k  f ðgÞ -  ∂g k N ð15Þ

~ is separable and compact under the Note that the vector space Ω metric ρðu; u0 Þ. Now, we consider the sequence fUðNÞ ðtÞ : t Z0g of infinite~ ). It dimensional Markov processes on state space ΩN (or Ω N indicates that the stochastic evolution of this supermarket model with impatient customers is described as the infinite-dimensional Markov process fUðNÞ ðtÞ; t Z 0g, where d ðNÞ U ðtÞ ¼ AN f ðUðNÞ ðtÞÞ; dt where AN acting on functions f : ΩN -R is the generating operator of the Markov process fUðNÞ ðtÞ; t Z 0g, Out AN ¼ AIn N þA N ;

ð16Þ

for g A ΩN

h  i 1 ek  d1 d1  f ðgÞ ; AIn N f ðgÞ ¼ λN ∑ ðg k  1  g k Þ f g þ N k¼1

1 ∂ Af ðgÞ ¼ λ ∑ ðg dk 1 1 g dk1 Þ f ðgÞ ∂g k k¼1 1 ∂  μ ∑ ½ð1 g k þ 1 Þd2  ð1  g k Þd2  f ðgÞ ∂g k k¼1 1

 θ ∑ ðk  1Þðg k  g k þ 1 Þ k¼2

∂ f ðgÞ: ∂g k

ð21Þ

ðtÞ Let uðtÞ ¼ limN-1 uðNÞ ðtÞ for t Z 0, where uk ðtÞ ¼ limN-1 uðNÞ k for k Z 0. Based on the limiting generating operator A given in (21), as N-1 it follows from the system of differential equations (5)– (8) that uðtÞ is a solution to the following system of differential equations n

o

d2 o d1 n d u1 ðtÞ ¼ λ ∑ ½u0 ðtÞd1  ½u1 ðtÞd1  μ ∑ ½1  u2 ðtÞd2  ½1  u1 ðtÞd2 ; dt m¼1 m¼1

and for k Z 2 o d1 n d2 n d u ðtÞ ¼ λ ∑ ½uk  1 ðtÞd1  ½uk ðtÞd1  μ ∑ ½1  uk þ 1 ðtÞd2 dt k m¼1 m¼1

1

½1  uk ðtÞd2 g ðk  1Þθ½uk ðtÞ uk þ 1 ðtÞ;

ð18Þ

where ek stands for a row vector with the kth entry 1 and all others 0. For g A ΩN , it follows from Eqs. (16) to (18) that h  i 1 e  AN f ðgÞ ¼ λN ∑ ðg dk1 1  g dk 1 Þ f g þ k  f ðgÞ N k¼1 h  i 1 e  d2 þ μN ∑ ½ð1  g k þ 1 Þ  ð1  g k Þd2  f g  k  f ðgÞ N k¼1 h  i 1 ek  þ θN ∑ ðk  1Þðg k g k þ 1 Þ f g  f ðgÞ : ð19Þ N k¼2 The operator semigroup of the Markov process fUðNÞ ðtÞ; t Z0g is defined as TN ðtÞ, where if f : ΩN -R, then for g A ΩN and t Z0 TN ðtÞf ðuÞ ¼ E½f ðUN ðtÞjUN ð0Þ ¼ g:

it follows from (19) that as N-1

ð22Þ ð17Þ

and h  i ek  d2 d2  f ðgÞ AOut N ¼ μN ∑ ½ð1  g k þ 1 Þ  ð1  g k Þ  f g  N k¼1 h  i 1 ek   f ðgÞ þ θN ∑ ðk  1Þðg k  g k þ 1 Þ f g  N k¼2

Now, we consider the limiting process fUðtÞ : t Z 0g of the sequence fUðNÞ ðtÞ : t Z 0g of Markov processes for N ¼ 1; 2; 3; …, that is, UðtÞ ¼ limN-1 UðNÞ ðtÞ for t Z 0, which leads to that uðtÞ ¼ limN-1 uðNÞ ðtÞ for t Z 0. In this case, two formal limits for the sequence fAN g of generating operators and for the sequence fTN ðtÞg of semigroups are expressed as A ¼ limN-1 AN and TðtÞ ¼ limN-1 TN ðtÞ for t Z0, respectively. Note that as N-1 h  i e  ∂ N f g þ k  f ðgÞ - f ðgÞ ∂g k N

ð20Þ

Note that AN is the generating operator of the operator semigroup TN ðtÞ, it is easy to see that TN ðtÞ ¼ expfAN tg for t Z0. Definition 1. A operator semigroup fSðtÞ : t Z0g on the Banach ~ Þ is said to be strongly continuous if lim space L ¼ CðΩ t-0 SðtÞf ¼ f for every f A L; it is said to be a contractive semigroup if J SðtÞ J r 1 for t Z 0.

ð23Þ

with the boundary condition u0 ðtÞ ¼ 1;

ð24Þ

and the initial condition uk ð0Þ ¼ g k ;

k Z 0;

ð25Þ

where 1 ¼ g 0 Z g 1 Zg 2 Zg 3 Z ⋯ Z 0: It is worthwhile to note that in the next section, we shall prove the existence and uniqueness of solution to the system of differential equations (22)–(25) by means of the Picard approximation. We define a mapping: g-uðt; gÞ, where uðt; gÞ is a solution to the system of differential equations (22)–(25). It is clear that uð0; gÞ ¼ g. For the operator semigroup TðtÞ acts in the space L. If ~ , then f A L and g A Ω TðtÞf ðgÞ ¼ f ðuðt; gÞÞ:

ð26Þ

~ -R with uniform metric J f J ¼ maxjf ðuÞj, and similarly, let f :Ω

It is easy to see that the operator semigroups TN ðtÞ and TðtÞ are strongly continuous and constructive. We denote by DðAÞ the domain of the generating operator A. It follows from (26) that if f is a function from L and has the partial derivatives ð∂=∂g k Þf ðgÞ A L for k Z 0, and supfð∂=∂g k Þf ðgÞg o 1, then f A DðAÞ.

~ induces a contraction mapping LN ¼ CðΩN Þ. The inclusion ΩN  Ω Π N : L-LN ; Π N f ðuÞ ¼ f ðuÞ for f A L and u A ΩN .

Let D be the set of all functions f A L that have the partial derivatives ð∂=∂g k Þf ðgÞ and ð∂2 =∂g i ∂g j Þf ðgÞ, and there exists a

~ Þ be the Banach space of continuous functions Let L ¼ CðΩ ~ uAΩ

kZ0

Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

positive constant C ¼ Cðf Þ o þ 1 such that 
  ∂  sup  f ðgÞ o C ∂g kZ0 k ~

 θðk  1Þðu0k;j  u0k þ 1;j Þ: ð27Þ

Using Lemma 2, we obtain Inequality (29) with a ¼ λd1 þ μd2 þ ðk  1Þθ; b0 ¼ 0; ϱ ¼ 1:

gAΩ

and ) (  ∂2    sup  f ðgÞ o C: ∂g i ∂g j  i;j Z 0

81

This completes this proof. ð28Þ

~ gAΩ

We call that f A L depends only on the first K variables if for ~ , it follows from g ð1Þ ¼ g ð2Þ for 0 r i r K  1 that gð1Þ ; gð2Þ A Ω i i ð1Þ f ðg Þ ¼ f ðgð2Þ Þ. Proposition 2 in Vvedenskaya et al. [50] indicates that the set of functions from L that depends on the first finite variables is dense in L. Definition 2. Let A be a closed linear operator on the Banach ~ Þ. A subspace D of DðAÞ is said to be a core for A if the space L ¼ CðΩ closure of the restriction of A to D is equal to A, i.e., AjD ¼ A. The following lemma comes from Proposition 1 in Vvedenskaya, Dobrushin and Karpelevich. We restated it here for convenience of description. Lemma 2. Consider an infinite-dimensional system of differential equations: For k Z 0, zk ð0Þ ¼ ck

□

Lemma 4. The set D is a core for the generating operator A. Proof. It is obvious that D is dense in L and D A DðAÞ. Let D0 be the set of functions from D which depend only on the first finite variables. It is easy to see that D0 is dense in L. Using Proposition 3.3 in Chapter 1 of Ethier and Kurtz [22], it can show that for any t Z 0, the operator semigroup TðtÞ does not bring D0 out of D. Select an arbitrary function φ A D0 and let f ðgÞ ¼ φðuðt; gÞÞ for ~ . It follows from Lemma 3 that f has the partial derivatives gAΩ ∂uk ðt; gÞ=∂g j and ∂2 uk ðt; gÞ=∂g i ∂g j and they satisfy the inequalities (29) and (30). Therefore f A D. This completes the proof. □ The following theorem applies the operator semigroup to provide an mean-field limit, which shows that the sequence fUðNÞ ðtÞ : t Z 0g of Markov processes asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of differential equations (23)–(25). ~ -R and t 4 0, Theorem 1. For any continuous function f: Ω lim sup jTN ðtÞf ðgÞ f ðuðt; gÞÞj ¼ 0;

N-1 g A Ω N

and

and the convergence is uniform in t within any bounded interval.

1 dzk ðtÞ ¼ ∑ zi ðtÞai;k ðtÞ þ bk ðtÞ; dt i¼0

∑1 i ¼ 0 jai;k ðtÞj r a; jbk ðtÞj r b0

and let a o b. Then

zk ðtÞ r ϱ expfat g þ

expfbtg; jck j r ϱ; b0 Z0

and

  b0 ½exp bt  expfat g: ba

The following lemma is a key to prove that the set D is a core for the generating operator A. Lemma 3. Let uðt; gÞ be a solution to the system of differential equations (22)–(25). Then ) (  ∂    sup  f ðgÞ rexpfat g ð29Þ ∂g k  kZ0 ~ gAΩ

and 8 9 = 2;k  1;k   1;k N ;  N  r C ; max  2 2   N > > N ∂g k ∂g k : N   ; we obtain   C   AN f ðgÞ  Af ðgÞ r N 1

(

1

1

∑ ðg dk 1 1  g dk1 Þ þ ∑ ½ð1 g k Þd2  ð1  g k þ 1 Þd2 

k¼1

k¼1

)

þ ∑ ðk 1Þðg k g k þ 1 Þ Proof. We only prove Inequality (29), while Inequality (30) can be proved similarly. It is easy to verify that the solution uðt; gÞ to the system of differential equations (22)–(25) possesses the partial derivatives ∂uk ðt; gÞ=∂g j and ∂2 uk ðt; gÞ=∂g i ∂g j for i; j Z 0. For simplicity of description, we will omit the argument of ∂2 uk ðt; gÞ=∂g i ∂g j . We write uk ¼ uk ðt; gÞ; u0k;j ¼

∂uk ðt; gÞ ∂g j

It follows from (23) that for all k; j Z 2, du0k;j dt

¼ d1 λðukd111 u0k  1;j udk 1  1 u0k;j Þ þ d2 μ½ð1 uk þ 1 Þd2  1 u0k þ 1;j  ð1  uk Þd2  1 u0k;j 

k¼2

1 C r ½g d01 þ ð1  g 1 Þd2 þ ∑ g k : N k¼1

because C and g d01 þ ð1 g 1 Þd2 þ∑1 k ¼ 1 g k are all finite (note that ∑1 k ¼ 1 g k is the expected queue length at time 0, hence ∑1 k ¼ 1 g k o þ1), it is clear that as N-1, lim sup jAN f ðgÞ  Af ðgÞj ¼ 0:

N-1 g A Ω N

This completes the proof.

□

Finally, we provide some interpretation on Theorem 1. If limN-1 UðNÞ ð0Þ ¼ uð0Þ ¼ g A Ω in probability, then Theorem 1 shows that UðtÞ ¼ limN-1 UðNÞ ðtÞ is concentrated on the trajectory

82

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Γ g ¼ fuðt; gÞ : t Z 0g. This indicates the functional strong law of large numbers for the time evolution of the fraction of each state of this supermarket model, thus the sequence fUðNÞ ðtÞ; t Z 0g of Markov processes converges weakly to the expected fraction vector uðt; gÞ as N-1, that is, for any T 40 lim

sup J UðNÞ ðsÞ  uðs; gÞ J ¼ 0 in probability:

ð31Þ

N-1 0 r s r T

Now, we provide two useful properties for the Gateaux derivatives of the infinite-dimensional functions. Obviously, the two useful properties also hold for the Frechet derivatives. 1 If the infinite-dimensional function G : R1 þ -R þ is Gateaux differentiable, then there exists a vector t ¼ ðt 1 ; t 2 ; t 3 ; …Þ with 0 r t k r 1 for k Z 1 such that GðyÞ  GðxÞ ¼ ðy  xÞDGðx þt⊘ðy  xÞÞ:

ð34Þ

Furthermore, we have J GðyÞ GðxÞ J r sup J DGðx þ tðy  xÞÞ J J y x J :

5. Existence and uniqueness In this section, we provide a computational framework for organizing a Lipschitzian condition for the infinite-dimensional 1 1 expected fraction vector function F : R 1 þ -C ðR þ Þ, which is given by the system of differential equations (22)–(25). Then we use the Lipschitzian condition to show the unique and global solution to the system of differential equations by means of the Picard approximation. To organize the Lipschitzian condition, we need to use the 1 derivative of the infinite-dimensional function G : R1 þ -R þ . Thus it is necessary to first provide some useful notation and definitions of derivatives. Readers may refer to Li et al. [29] for more details. 1 For the infinite-dimensional function G : R 1 þ -R þ , we write x ¼ ðx1 ; x2 ; x3 ; …Þ and GðxÞ ¼ ðG1 ðxÞ; G2 ðxÞ; G3 ðxÞ; …Þ, where xk and Gk(x) are scalar for k Z1. Then the matrix of partial derivatives of the infinite-dimensional function G(x) is defined as 0 ∂G ðxÞ ∂G ðxÞ ∂G ðxÞ 1 1 2 3 ⋯ ∂x1 ∂x1 ∂x1 B C B ∂G1 ðxÞ ∂G2 ðxÞ ∂G3 ðxÞ ⋯ C B C ∂x2 ∂x2 ð32Þ DGðxÞ ¼ B ∂x2 C: B ∂G1 ðxÞ ∂G2 ðxÞ ∂G3 ðxÞ ⋯ C @ ∂x3 A ∂x3 ∂x3 ⋮ ⋮ ⋮ Now, we define two classes of derivatives for the infinitedimensional function G(x). In fact, they are a direct and minor generalization from the derivatives of finite-dimensional functions. 1 Definition 3. Let the infinite-dimensional function G : R 1 þ -R þ . 1 (1) If there exists a linear operator A : R1 þ -R þ such that for any 1 vector hA R and a scalar t A R

lim t-0

ð35Þ

0rtr1

J Gðx þ thÞ  GðxÞ thA J ¼ 0; t

then the function GðxÞ is called to be Gateaux differentiable at 0 x A R1 þ . In this case, we write the Gateaux derivative GG ðxÞ ¼ A. 1 (2) If there exists a linear operator B : R 1 -R such that for any þ þ vector hA R1

For simplicity of description, we write that uk ¼ uk ðtÞ for k Z 0. In this case, the system of differential equations (22)–(25) can be simplified as an initial value problem as follows: d u1 ¼ λðud01  ud11 Þ  μ½ð1  u2 Þd2 ð1 u1 Þd2 ; dt

ð36Þ

for k Z 2, d u ¼ λðudk 1 1  udk1 Þ  μ½ð1  uk þ 1 Þd2 ð1  uk Þd2   θðk  1Þðuk  uk þ 1 Þ; dt k ð37Þ with the boundary condition u0 ¼ 1;

ð38Þ

and the initial condition uk ð0Þ ¼ g k ;

k Z 1:

ð39Þ

Let x ¼ ðx1 ; x2 ; x3 ; …Þ ¼ ðu1 ; u2 ; u3 ; …Þ. Similarly, set FðxÞ ¼ ðF 1 ðxÞ ; F 2 ðxÞ; F 3 ðxÞ; …Þ. From (36) to (39), we write F 1 ðxÞ ¼ λð1  xd11 Þ  μ½ð1  x2 Þd2  ð1  x1 Þd2 ; for k Z 2 F k ðxÞ ¼ λðxdk 1 1 xdk 1 Þ  μ½ð1  xk þ 1 Þd2  ð1  xk Þd2   θðk  1Þðxk  xk þ 1 Þ: ð40Þ 2

ðR1 þ Þ.

It is clear that F(x) is in C In what follows we show that the infinite-dimensional function F(x) is Lipschitzian for t Z 0. From (1) of Definition 3 and (33), the matrix of partial derivatives of the function F(x) is given by 0 1 A1 ðxÞ B2 ðxÞ B C ðxÞ A ðxÞ B ðxÞ C 2 3 B 1 C DFðxÞ ¼ B ð41Þ C; @ A C 2 ðxÞ A3 ðxÞ B4 ðxÞ ⋱

⋱

⋱

where A1 ðxÞ ¼  λd1 xd11  1  μd2 ð1  x1 Þd2  1 ; C 1 ðxÞ ¼ μd2 ð1  x2 Þd2  1 ;

J Gðx þ hÞ  GðxÞ hB J ¼ 0; JhJ J h J -0 lim

and for k Z 2

then the function GðxÞ is called to be Frechet differentiable at 0 x A R1 þ . In this case, we write the Frechet derivative GF ðxÞ ¼ B.

Ak ðxÞ ¼  λd1 xkd1  1  μd2 ð1  xk Þd2  1  ðk  1Þθ; Bk ðxÞ ¼ λd1 xdk 111 ;

It is easy to check that if the infinite-dimensional function GðxÞ is Frechet differentiable, then it is also Gateaux differentiable. At the same time, we have G0G ðxÞ ¼ G0F ðxÞ ¼ DGðxÞ:

ð33Þ

Let t ¼ ðt 1 ; t 2 ; t 3 ; …Þ with 0 rt k r1 for k Z1. We write 0 ∂G ðx þ t ðy  xÞÞ ∂G ðx þ t ðy  xÞÞ ∂G ðx þ t ðy  xÞÞ 1

1

∂x1 B B ∂G1 ðx þ t1 ðy  xÞÞ B ∂x2 DGðx þ t⊘ðy  xÞÞ ¼ B B ∂G1 ðx þ t1 ðy  xÞÞ @ ∂x3 ⋮

2

2

∂x1 ∂G2 ðx þ t 2 ðy  xÞÞ ∂x2 ∂G2 ðx þ t 2 ðy  xÞÞ ∂x3

⋮

3

3

∂x1 ∂G3 ðx þ t 3 ðy  xÞÞ ∂x2 ∂G3 ðx þ t 3 ðy  xÞÞ ∂x3

⋮

C k ðxÞ ¼ μd2 ð1 xk þ 1 Þd2  1 þ ðk  1Þθ: Hence it follows from (41) that (

)

J DFðxÞ J ¼ max J A1 ðxÞ þ C 1 ðxÞ J ; sup J Bk ðxÞ þ Ak ðxÞ þ C k ðxÞ J : ⋯

kZ2

1

C ⋯C C C: ⋯C A

Note that J A1 ðxÞ þ C 1 ðxÞ J o μd2 ;

ð42Þ

and for k Z 2 J Bk ðxÞ þ Ak ðxÞ þ C k ðxÞ J o λd1 þ μd2 ;

ð43Þ

Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

83

it follows from Equations (42) and (43) that

It follows from (48) that

J DFðxÞ J r λd1 þ μd2 ¼ M;

λπ dk1  μ½1  ð1  π k þ 1 Þd2   θ ∑ π i ¼ 0;

1

def

~ Note that x ¼ u, this gives that for u A Ω J DFðuÞ J r M:

i ¼ kþ1

ð44Þ

~ It follows from (35) and (44) that for u; v A Ω J FðuÞ FðvÞ J r M J ðu  vÞ J :

ð45Þ

~. This indicates that the function FðuÞ is Lipschitzian for u A Ω Note that x ¼ u and uð0Þ ¼ g, it follows from Equations (22) to ~ (25) that for u A Ω Z t uðtÞ ¼ uð0Þ þ FðuðsÞÞ ds; 0

this gives Z

t

uðtÞ ¼ g þ

ð46Þ

FðuðsÞÞ ds: 0

Based on the integral equation (46), we can apply the Picard approximation to prove the existence and uniqueness of solution to the system of differential equations (22)–(25), where the Lipschitzian condition organized above is a key when using the Picard approximation. At the same time, it is worthwhile to note that once the Lipschitzian condition is determined by our above method, the existence and uniqueness of solution can be proved by a standard method in the Banach spaces. Readers may refer to Li et al. [29] for more details.

this gives that for k Z2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

πk ¼

d1

ρ  1 ½1 ð1  π k þ 1 Þd2  þ

θ 1 ∑ π: λ i ¼ kþ1 i

ð52Þ

pffiffiffiffiffiffiffiffiffiffiffi θ ¼0, then π 1 ¼ 1  d2 1  ρ and 2. If qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 d1 π k þ 1 ¼ 1  1  ρπ k for k Z 1. In this case, it is easy to compute Remark

the fixed point π ¼ ðπ 0 ; π 1 ; π 2 ; …Þ for various values of d1 ; d2 Z 1. At the same time, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i d

∑id2¼01 2 1  ρπ dk 1 1 πk þ 1 πk : ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 

πk

∑di 2¼11

d2

πk  1

1  ρπ dk 1

This product decomposition shows the associated role played by the two choice numbers d1 and d2, respectively. It is observed from (51) and (52) that the parameter θ related to the impatient customers makes computation of the fixed point more difficult. In this case, it follows from (47) that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π 2 ¼ 1  d2 ρð1  π d11 Þ þ ð1  π 1 Þd2 ð53Þ and from (48) that 1

6. The fixed point

λπ dk1 1  μ½1  ð1  π k Þd2   θ ∑ π i ¼ 0

In this section, we study the fixed point of the infinitedimensional system of differential equations (22)–(25), and provide an effective and efficient algorithm for computing the fixed point by means of an infinite-dimensional system of nonlinear equations. Note that the fixed point is a key for performance analysis of this supermarket model with impatient customers. Let π ¼ ðπ 0 ; π 1 ; π 2 ; π 3 ; …Þ. A row vector π is called a fixed point of the infinite-dimensional system of differential equations (22)– (25) satisfied by the limiting expected fraction vector uðtÞ, if π ¼ limt- þ 1 uðtÞ, or π k ¼ limt- þ 1 uk ðtÞ for k Z 0. It is well-known that if π is a fixed point of the expected fraction vector uðtÞ, then

d uðtÞ ¼ 0: lim t- þ 1 dt Taking t- þ1 in both sides of the system of differential equations (22)–(25), we obtain

λðπ d01  π d11 Þ  μ½ð1  π 2 Þd2  ð1  π 1 Þd2  ¼ 0;

ð47Þ

for k Z 2,

λðπ dk1 1  π dk1 Þ  μ½ð1  π k þ 1 Þd2  ð1  π k Þd2   θðk  1Þðπ k  π k þ 1 Þ ¼ 0; ð48Þ with the boundary condition

π 0 ¼ 1:

ð49Þ

In what follows we show how to solve the system of nonlinear equations (47)–(49). From (47) and (48), we get 1

λπ d01  μ½1 ð1  π 1 Þd2   θ ∑ π k ¼ 0;

ð50Þ

k¼2

which follows sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

π1 ¼ 1 

d2

1

1  ρ þ θ ∑ πk: k¼2

i¼k

and 1

λπ dk1  μ½1  ð1  π k þ 1 Þd2   θ ∑ π i ¼ 0; i ¼ kþ1

they give

λðπ dk1 1  π dk1 Þ  μ½ð1  π k þ 1 Þd2  ð1  π k Þd2   θπ k ¼ 0: Hence we obtain that for k Z1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π k þ 1 ¼ 1  d2 ρðπ dk1 1  π dk1 Þ þ ð1  π k Þd2  θπ k :

ð54Þ

This gives some useful iterative relations for computing the fixed point once π1 can first be determined. Now, we further discuss the computational role played by the iterative relations (53) and (54). Let π 1 ¼ g A ð0; 1Þ. Then it is easy to see from the iterative relations (53) and (54) that πk is a function of the underdetermined number g, denoted as π k ðgÞ. From (50), it is shown that g is the maximal positive solution in (0, 1) to the nonlinear equation 1

λπ d01  μ½1  ð1  π 1 Þd2   θ ∑ π k ¼ 0; k¼2

this gives 1

λ  μ½1  ð1  gÞd2   θ ∑ π k ðgÞ ¼ 0:

ð55Þ

k¼2

In general, πk always decreases very fast for d1 ; d2 Z 2, as k increases. In this case, we take N as a suitable positive integer such that π N þ j  0 for jZ 1. Therefore, g can be approximately solved from the nonlinear equation N

λ  μ½1  ð1  gÞd2   θ ∑ π k ðgÞ ¼ 0: k¼2

ð51Þ Specifically, if

θ ¼ 0, then g ¼ 1 

pffiffiffiffiffiffiffiffiffiffiffi 1  ρ.

d2

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Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

Summarizing the above analysis, the following theorem provides an iterative algorithm for computing the fixed point π ¼ ðπ 0 ; π 1 ; π 2 ; …Þ, while its proof is easy and is omitted here. Theorem 2. If g is approximately solved from the nonlinear equation N

λ  μ½1  ð1  gÞd2   θ ∑ π k ðgÞ ¼ 0;

ð56Þ

k¼2

then

π 0 ¼ 1; π1 ¼ g and for k Z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π k þ 1 ¼ 1  d2 ρðπ dk1 1  π dk1 Þ þ ð1  π k Þd2  θπ k :

Using Theorem 2, we consider some special cases as follows: (1) If

θ ¼0 and d2 ¼ 1, then k

π k ¼ ρðd1  1Þ=ðd1  1Þ ;

performance approximation of this supermarket model with a bigger number N. Here, we omit its proof, while the proof can be completed by a similar discussion to those of Theorems 1 (iii) and Theorem 4 in Martin and Suhov [35]. Theorem 3. (1) If ρ o 1, then for any g A Ω we have lim uðt; gÞ ¼ π :

t- þ 1

Furthermore, there exists a unique probability measure φ on Ω, which is invariant under the map g⟼uðt; gÞ, that is, for any continuous function f : Ω-R and t 4 0 Z Z f ðgÞ dφðgÞ ¼ f ðuðt; gÞÞ dφðgÞ: Ω

Ω

Also, φ ¼ δπ is the probability measure concentrated at the fixed point π. (2) If ρ o 1, then for a fixed number N ¼ 1; 2; 3; …, the Markov process fUðNÞ ðtÞ; t Z0g is positive recurrent, and hence it has a unique invariant distribution φN . Furthermore, fφN g weakly converges to δπ , that is, for any continuous function f : Ω-R lim EφN ½f ðgÞ ¼ f ðπ Þ:

N-1

k Z 0;

where the fixed point fπ k g is called to be doubly exponential. (2) If θ 40 and d2 ¼ 1, then

Based on Theorems 3, we obtain a useful relation as follows: lim

π 2 ¼ π 1  ρð1  π d11 Þ

lim uðNÞ ðt; gÞ ¼ lim

t- þ 1 N-1

and for k Z 2

lim uðNÞ ðt; gÞ ¼ π :

N-1 t- þ 1

Therefore, we have

π k þ 1 ¼ ð1 þ θÞπ k þ ρðπ

d1  k1

π

d1 Þ: k

lim uðNÞ ðt; gÞ ¼ π :

N-1 t- þ 1

This gives 1

∑ πk ¼

k¼2

ρ  π1 ρ  g ¼ : θ θ

Using (56), we have N

λ  μg  θ ∑ π k ¼ 0; k¼2

this give π 1 ¼ g ¼ ρ. pffiffiffiffiffiffiffiffiffiffiffi (3) If θ ¼0 and d1 ; d2 Z 2, then π 1 ¼ 1  d2 1  ρ and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π k ¼ 1  d2 1  ρπ dk1 1 for k Z 2. At the same time, it is clear that lim

πk

ρ

k k-1 ðd1  1Þ=ðd1  1Þ

¼0

or ln π −1 k

lim

k→∞ðdk −1Þ=ðd −1Þ 1 1

¼ 0;

where the fixed point fπ k g is called to be super-exponential. (4) If θ 40 and d1 ; d2 Z 2, then lim

π k ðθ 4 0Þ ¼0 π θ ¼ 0Þ

k-1 k ð

and lim

π k ðθ ¼ 0Þ ρ

k

k-1 ðd1  1Þ=ðd1  1Þ

¼ 0;

where the fixed point fπ k g is also called to be superexponential. In the remainder of this section, we discuss some useful convergent properties of uðNÞ ðt; gÞ as N-1 and/or t-0. Under the condition ρ o 1, the following theorem describes two important limiting processes which are related to the fixed point. The two limiting processes are crucial for understanding the convergence of uðNÞ ðt; gÞ as N-1 and/or t-0, and specifically, for

7. Performance analysis In this section, based on the fixed point given in Theorem 2, we provide performance analysis of this supermarket model with impatient customers. Also, we use some numerical examples to discuss how the performance measures depend on some crucial factors of this supermarket model. Specifically, we observe that for improving system performance, the SLQ(d2) controlling the service processes is more effective than the JSQ(d1) controlling the arrival processes. Now, we provide two useful performance measures of this supermarket model with impatient customers: The mean of stationary queue length in any waiting line, and the expected sojourn time of any arriving customer spending in this system. Let Q be the stationary queue length of any waiting line (including the customer being served in the server, if any). Then 1

1

k¼1

k¼1

E½Q ¼ ∑ PfQ Z kg ¼ ∑ π k :

ð57Þ

If ρ o 1, then this supermarket model is stable. In this case, we denote by S the sojourn time of any arriving customer spending in this system. Note that a customer may leave the system because either it is impatient or after it is served, we need to introduce some events I ¼ fThe arriving customer shall leave the system because it is impatientg; for k Z 1 Ik ¼ fThe arriving customer finds k customers in its selected waiting line; and it shall leave the system because it is impatientg; J ¼ fThe arriving customer shall leave the system because it is servedg; for l Z 0

Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

Jl ¼ fThe arriving customer finds k customers in its selected waiting

E½S ¼

line; and it shall leave the system because it is servedg: It is easy to see that when computing E½S, the sample space is given by

1

1

I ¼ ⋃ Ik ;

J ¼ ⋃ Jl :

k¼1

l¼0

Based on the total probability theorem, we obtain E½S ¼ E½S; I þE½S; J; where 1

1

k¼1

k¼1

E½S; I ¼ ∑ E½S; Ik  ¼ ∑ E½SjIk PfIk g and 1

1

l¼0

l¼0

E½S; J ¼ ∑ E½S; Jl  ¼ ∑ E½SjJl PfJl g: Since the probability that an arriving customer finds k customers in the selected waiting line is given by π dk1  π dk 1þ 1 for k Z 0, we obtain

"

μ

þ

Λ ¼ I⋃J; where

1

ðπ  π

1

d1 0

ð60Þ

d1 =1,d2 =1

)

k

0.3

i¼1

0.25

E[Q]

 lþ1 E½S Jl  ¼

μ

and

(

where X is the exponential impatient time of impatience rate θ, and Yi is the exponential service time of the ith customer. This gives ( ) k 1 1 E½S; I ¼ ∑ ðπ dk1  π dk 1þ 1 ÞP X r ∑ Y i ð58Þ i¼1

kþ1

1

∑

μ

k¼1

( ðπ  π d1 k

d1 ÞP kþ1

k

i¼1

þ1

"

)

k

∑ Y i oy

P

i¼1

0

¼θ Let

(

þ1

Z

þ1

"

X 4 ∑ Yi :

k1 ð

e  θy 1  ∑

k1 ð

e  θy 1  ∑

j¼0

μyÞj j!

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

μyÞj j!

=2, =1

ð59Þ

1.25

i¼1

  dP X r y

j¼0

0

0.2

)

It is easy to see that ∑ki ¼ 1 Y i is the sum of k exponential service times, and it is the Erlang distribution of order k, given by ( ) k k  1 ðμyÞj P ∑ Yi ry ¼ 1 ∑ e  μy : i¼1 j ¼ 0 j! Hence we obtain ( ) Z k P X 4 ∑ Yi ¼

d1 =6,d2 =5

0.05

d1 =1,d2 =1 d1 =2,d2 =3 d1 =3,d2 =2

1.2

d1 =5,d2 =6 d1 =6,d2 =5

E[S]

E½S; J ¼ ðπ  π

d1 1 Þþ

d1 =5,d2 =6

Fig. 3. The mean of stationary queue length vs. λ for the different values of d1 ; d2 .

and d1 0

d1 =3,d2 =2

0.1

i¼1

θk¼1

d1 =2,d2 =3

0.15

)

l

PfJl g ¼ ðπ dl 1  π dl þ1 1 ÞP X 4 ∑ Y i ;

0

1

∑ ð1  ωk Þðπ dk1  π dk 1þ 1 Þ:

=2, =1

PfIk g ¼ ðπ dk 1  π dk1þ 1 ÞP X r ∑ Y i ;

Z

k¼1

d1 Þ kþ1

0.35

(

ωk ¼ θ

∑ ωk ðk þ1Þðπ  π d1 k

θk¼1

θ

μ

#

1

d1 1 Þþ

In the remainder of this section, using (57) and (60) we provide some numerical examples to analyze how the two performance measures E½Q and E½S depend on some crucial parameters of this supermarket model, for example, d1 ; d2 ; λ; μ and θ. Example one: the role of the arrival process: We consider the supermarket model with impatient customers, where the Poisson arrival rate λ A ½0:1; 0:9, the exponential service rate μ ¼1 and the exponential impatience rate θ ¼ 2. Figs. 3 and 4 respectively indicate how E½Q and E½S depend on the arrival rate λ A ½0:1; 0:9 under the different values: d1 ; d2 ¼ 1; 2; 3; 5; 6. It is seen from Figs. 3 and 4 that E½Q and E½S increase as λ increases, while each of them decreases as d1 and d2 increase. At the same time, it is easy to see that for improving system performance, the choice number d2 for controlling the service process is more effective than the choice number d1 for controlling the arrival process.

 1 E½S Ik  ¼ ;

1

85

1.15

1.1

# e  μy dy:

1.05

# e  μy dy:

Then it follows from (58) and (59) that

1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 4. The expected sojourn time vs. λ for the different values of d1 ; d2 .

0.9

86

Q.-L. Li et al. / Computers & Operations Research 55 (2015) 76–87

=0.5, =1 0.7

d1=1,d2=1 d1=2,d2=3

0.6

d1=3,d2=2 d1=5,d2=6

E[Q]

0.5

d1=6,d2=5

0.4

0.3

0.2

0.1

0 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 5. The mean of stationary queue length vs. θ for the different values of d1 ; d2 .

=0.5, =1 1.45

d1=1,d2=1 d1=2,d2=3

1.4

d1=3,d2=2

1.35

d1=5,d2=6 d1=6,d2=5

E[S]

1.3 1.25 1.2

doubly dynamic control, which may also be related to the sizebased scheduling with respect to the arrival and service processes. Firstly, we provide a probability method to set up the infinitedimensional system of differential equations. Then we use the operator semigroup to provide the mean-field limit, which shows that the sequence of infinite-dimensional Markov processes asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Finally, we provide an effective and efficient algorithm for computing the fixed point of the infinitedimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model. Also, some numerical examples are given to demonstrate how the performance measures depend on some crucial parameters of this supermarket model. From many practical applications, analysis of such a supermarket model is an important parallel queueing network to analyze the relation between the system performance and the job routing rule, and it can also help to design reasonable architecture to improve the performance and to balance the load. Note that this paper provide a clear picture for how to use the mean-field theory as well as the numerical computation to analyze performance measures of more general supermarket models. We show that this picture is organized as three key parts: (1) setting up system of differential equations, (2) giving system of nonlinear equations satisfied by the fixed point through the mean-field limit, and (3) performance numerical computation of this supermarket model. Therefore, the results of this paper give new highlight on understanding influence of resource scheduling and information utilization on performance measures of more general supermarket models. Along such a line, there are a number of interesting directions for potential future research, for example:

 analyzing non-Poisson inputs such as renewal processes;  studying non-exponential service time distributions, for exam-

1.15 1.1



1.05 1 0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ple, general distributions, matrix-exponential distributions and heavy-tailed distributions; and discussing the bulk arrival processes, and/or the bulk service processes, where effective algorithms both for the fixed point and for the performance measures are necessary and interesting.

Fig. 6. The expected sojourn time vs. θ for the different values of d1 ; d2 .

Example two: the role of the impatient behavior: We consider the supermarket model with impatient customers, where the Poisson arrival rate λ ¼0.5, the exponential service rate μ ¼1 and the exponential impatience rate θ A ½0:4; 2. Figs. 5 and 6 respectively show how E½Q and E½S depend on the impatience rate θ A ð0:4; 2Þ under the different values: d1 ; d2 ¼ 1; 2; 3; 5; 6. It is easy to see that E½Q and E½S decrease as θ increases, while each of them decreases as d1 and d2 increase. At the same time, it is easy to see that for improving system performance, the choice number d2 for controlling the service process is more effective than the choice number d1 for controlling the arrival process. One interesting observation: It is seen from Figs. 3 to 6 that for improving system performance, the choice number d2 for controlling the service process is more effective than the choice number d1 for controlling the arrival process. However, we cannot give a strict proof for such a valuable observation yet. This may be an interesting topic in the future study.

8. Concluding remarks In this paper, we apply the mean-field theory to generalize the supermarket model both from the impatient customers and from a

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