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International Journal of Operations Research Vol. 3, No. 1, 1-6 (2006)

Optimal Selection of Arrival and Service Rates in Tandem Queues

Hsing Luh

1, *

and M.S. Moustafa

2

1Department of Mathematical Sciences, National ChengChi University, Taipei 116, Taiwan, R.O.C. 2Department of Science, The American University in Cairo, Egypt

Received June 2004; Revised September 2005; Accepted October 2005

Abstract

¾

We consider n M/M/1 queues in series. At queue one the arrival and service rates are chosen in pair from a finite set whenever there are arrivals or service completions at any queue. Customers arriving to queue k (k = 1, 2, …, n - 1) must go on to queue k + 1 after finishing service at server associated queue k. Customers arriving to queue n leave the system after finishing service at the last server. Arrival and service rates are fixed at queues 2 to n. The objective is to minimize the expected discounted cost of the system over finite and infinite horizons. We show that there is a monotone hysteretic optimal policy in which the arrival and service rates are decreasing and increasing, respectively, in the queue length. In order to establish the result, we formulate the optimal control problem with an equivalent Linear Programming. We believe that many optimal control queueing problems, in which the dynamic programming formulation fails, can be treated successfully via Linear Programming techniques.

Keywords

¾

Queueing networks, Markov decision processes, Stochastic linear programming, Sample-path arguments

*Corresponding author’s email: slu@nccu.edu.tw 1813-713X copyright © 2006 ORSTW

1. INTRODUCTION

In last decade, many authors considered the optimal control of systems with more than one server. Rosberg, Varaiya and Walrand (1982) considered two M/M/1 service stations in tandem with uncontrollable arrival process and control of service rate m Î[0, ]a at first station only. They have showed that the optimal policy is characterized by switching curves. A generalization of the model given by Rosberg, Varaiya, and Walrand (1982) for controlling the service rates in a cycle of m queues has been considered by Weber and Stidham (1987). They also studied the control of arrivals to the first queue only and service rates at each queue of m queues in series.

Hajek (1984) has considered a general two-node model. Nodes 1 and 2 have Poisson arrivals at rates λ1 and λ2

respectively. A third stream of Poisson arrivals at rate λ can be routed to either queue. The nodes have fixed exponential service rates μ1 and μ2 respectively. There are

two additional exponential servers, with rates v12 and v21.

The first of which serves queue 1 and sends jobs to queue 2. The second of which serves queue 2 and sends jobs to queue 1. Service completions by these servers can be “accepted” or “rejected”. The jobs arriving at rate λ must be routed to one of the two nodes. All decisions are made dynamically as a function of the number of jobs in the two queues. Hajek (1984) uses an inductive proof to establish the existence of a monotonic switching curve.

Ghoneim and Stidham (1985) studied two exponential servers in series (with mean service rates μ1 and μ2), each

with an infinite capacity queue. Arrivals to queue i are from

a Poisson process with mean rate λi, i = 1, 2. Jobs arriving

to queue 1 must go on to queue 2 after finishing service at server 1. Jobs arriving to queue 2 leave the system after finishing service at server 2. They have showed that λi are

nonincreasing in the number of customers in either queue. Moustafa (1992) considered two M/M/1 queues in series. At queue 1, the arrival and service rates are chosen in pair from a finite set. He showed numerically that the optimal policy is characterized by a switching curve, but he could not apply the induction proof to construct this structure of the optimal policy. The model considered by Moustafa (1992) is two node version of the model studied by Lu and Serfozo (1984). They used an inductive proof to show that there is a monotone hysteric optimal policy in which the arrival and service rates are decreasing and increasing respectively in queue length. A monotone hysteretic optimal policy defines a set of optimal policies which are specified by a range of queue sizes but its bounds are different when the number of customers is increasing or decreasing. The “hysteresis” refer to decisions that depend both on the current queue length and the current rates. For example, assuming there are only two pairs of arrival and service rates, it is optimal to choose the first pair of arrival and service rates when the queue size decreases to 7 from 10, but to choose the second pair when the queue size increases to 4 from 0. It mean that the bounds of the optimal policy are different when the number of customers is increasing or decreasing.

Typically, the control problem is formulated as a Markovian decision process and the tool of Dynamic

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Programming is used to establish the structure of the optimal policy. However, arguments based on Linear Programming (see Luh and Rieder, 2001) may be used in models where Dynamic Programming technique fails. Bertsimas and Nino% -Mora (1999) presented new lower bounds on the achievable cost that emerge as the values of

nonlinear programming problems over relaxed

formulations of the system’s achievable performance region.

In this paper, we use Linear Programming (LP) arguments to generalize the model considered by Moustafa (1992) and establish the structure of the optimal policy. We show that there is a monotone hysteretic optimal policy in which the arrival and service rates are decreasing and

increasing, respectively, in the queue length. We formulate the optimal control problem with an equivalent Linear Programming, exploring the structure of optimal policies by studying its dual solutions. Our approaches extend Lu and Serfozo’s (1984) results to more general cases and provide powerful new methodologies to optimal load distribution across a network of interconnected stations.

The paper is organized as follows: In section 2 we describe the queueing model of the system. In section 3 we provide the Linear Programming formulation of the optimal control problem. In section 4 we discuss the structure of the optimal policy. Finally, conclusion and future research are given in section 5.

Figure 1. A series of queues in tandem where ( , )l mi i is selected at station 1.

2. THE SYSTEM MODEL

We consider n M/M/1 queues in series as depicted in Figure 1. Customers arriving to queue k (k = 1, 2, …, n-1) must go on to queue k + 1 after finishing service at server

k. Customers arriving to queue n leave the system after

finishing service at the last server. At the first queue, the arrival and service rates are chosen in pair ( , )l mi i from

a finite set f={( , ),( , ),...,(l m1 1 l m2 2 l mm, m)} with

1 m 0

l ³L³l > and 0 <m1£L£mm.

A switching cost si occurs when the i-th pair is

substituted by another pair. We assume that si is increasing

of i and λ1 + μ1, …, λm + μm are positive but not identical.

At queues 2 to n, the service rates are fixed to vj, j = 2,

3, …, n. We consider selection decisions that are stationary and nonidling which means decisions have to be made at each decision point. Thus, the rates may be adjusted at decision epoch whenever the system state changes.

Our objective is to minimize the expected holding cost over finite and infinite horizons. Let l

k

x be the number of customers at queue l, l = 1, 2, …, n, when the k-th transitions (i.e., arrival or service completion) occurs. However, since there exists m classes of customers at queue 1, l

k

x is a total number of customers with different classes. Now, with abusing a bit of notation, let

i k

x denote the number of customers at queue 1 for

different class i, i = 1, 2, …, m but denote the number of customers at queue i

k

x for i = m + 1, m + 2, …, m + n -

1. For finite horizon N, we consider the discounting factor 0 < β < 1, the objective is given by

-1 0 1 min N km n ( i) f i k k i G E x f b a + = = =

å å

(1)

where αi (i = 1, 2,…, m + n) are the holding costs per

customer with respect to each i. For infinite horizon, we consider -1 1 0 1 lim(1 )min km n ( i) i i k k i G E x b b f b a ¥ + ® = = ì ü = - í ý î

å å

þ (2) We can use success approximations in (1) and establish these properties for both (1) and (2). It follows from the theory of Markov decision processes that Gf → Gi as N →

∞ and β → 1. See, e.g., Puterman (1994). In this paper, we only consider the case of αi = 1 for all i to concentrate on

the methodology itself.

3. LINEAR PROGRAMMING FORMULATION Now, we describe the LP formulation whose construction is sampled on system at the times corresponding to either arrivals, real service completions, or virtual service completions. Let

{

A A1, 2,...,A D Dm, 1, 2,...,D Um, m+1,Um+2,...,Um n+ -1

}

W =

be the set of all transitions. Here Ai and Di (i = 1, 2, …, m)

represent the arrival and departure respectively at queue 1 as the i-th pair of arrival and service rates are chosen. For j = m + 1, m + 2, …, m + n - 1, Uj represent the event of

departure at queue j - m + 1. Because we assume no idle between departure at queue j - 1 and the arrival at queue j, the event of departure at queue j - 1 is equivalent to arrival at queue j. Thus the event of arrival at queue j is not included in Ω.

Let w ÎWk represent the k-th transition of the

queueing system. Denote by W = W´ W´ ´W the k .... sample space of the system having k transitions,

i

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1 k N£ £ . Define wk=w w w1 2... kÎWk as k transitions

and Pr(wk) as the probability distribution over W . Let k

( )

i k k

x w represent the change in the system state incurred by transition wk with 1£ £ + - . If i mi m n 1 £ , then

the transition is either Ai or Di, which denotes the change

at queue 1. If i > m, then it denotes the state change at queue i - m + 1. The function i( k)

k

x w is given by for i = 1, 2, …, m (at the first queue)

1 ( ) 1 1, 2, ..., 0 . . k i i k k k i A D i m o w w x w w = ì ï = -í = = ï î

It is given at queue 2 that

1 1 ( ) 1 1, 2, ..., 0 . . k i m i k k k m D U i m o w w x + w w + = ì ï = -í = = ï î

and for queue i - m+1, m + 1 < i < m + n,

1 1 ( ) 1 0 . . k i i k k k i U U o w w w w -= ì ï x = -í = ï î Let i k

z (wk-1wk) denote the control variable that represents

selection of (λi,μi) taken at the k-th transition instant, i = 1,

2, …, m. We define

( ) {0,1}, , 1, 2, ..., .

i k i

k k j k j

z w Î w =A or w =D j= m

When Aj or Dj occurs we set z wki( k)= 1 if the i-th pair is

chosen at queue 1. For other queues, it is assigned to 1 since we did not specify any control action on transitions at queue l (l = 2, 3, …, n). This shall be explained in detail in next section. The evolution of the system is described by the following equation:

1 1 1 1( ) ( ) 1( ) 1( ), i k i k i k i k k k k k x w + x w x w + z w + + = + + 1, 2,..., 1. i = m n+

-Note both state variables x and control variables z are integer-valued. Principally, we shall construct an integer programming for this model. Instead, we derive a relaxed Linear Programming problem where all integer requirements are relaxed. The reason may be found in Luh and Viniotis (2002) in which the optimal solution of a similar formulation has been proved integer-valued. Suppose that the initial queue length is x0, we may rewrite

the cost incurred by policy z as

1 1 0 1 Pr( ) { ( )} k k N m n k i k k k w i w bk x w - + -= ÎW =

å å

å

1 0 0 (1 ... N)N i i x b b -= = + + +

å

1 1 1 0 ( ) ( ) k k N m n i k i k k k k w i w z w g - + -= ÎW = +

å å å

1 1 ( ) { Pr( ,..., ) ( ) } k j N i k k i k i j k j k j k w w w w w w s g x b + -= × =

å å

+ (3)

and z must be satisfied with the following three constraints.

(i) Nonnegative queue sizes

For every queue, we have nonnegative queue size constraints as follows. 0 i( k, ) k x w z £ =D 0 1 ( ) ( ) k i i j i j j j j x x w z w = +

å

(4) for all i, j, and k.

(ii) Non-idling policy

A non-idling policy means that the server should not be idled whenever the associated queue is not empty. Namely, for all i, 1 if (i k) 0 then i ( k ) 0 k k i x w > z + w U > (5) (iii) Assignment

For DrÎΩ at queue 1, r = 1, 2, …, m, set

1 1 ( ) 1 m i j j r i z w D -= =

å

(6) Since the initial queue sizes are given as parameters in the formulation, constraints (4) may be rearranged to place variables z on the left hand side and 0

i

x on the right hand

side of the inequality. Under the stability condition λi < μ, i

= 1, 2, …, m, the queue sizes never blow up in a finite time. Therefore, it is easy to check constraints (4)-(6) are bounded and feasible.

Lemma 3.1 The solution set of constraints (4)-(6) is consistent, and

the optimal solution exists.

Proof. Because N < ∞ and (3) is piecewise linear which was proved in Luh and Viniotis (2002), it is immediately clear.

Lemma 3.2 The optimal solutions of minimizing (3) subjecting to

(4)-(6) are integer valued.

Proof. The proof may be found in Luh and Viniotis (2002). Therefore, we omit it here.

Lemma 3.3 The optimal solutions 1i( )

r

z D correspond to the optimal policies which assign customers to queue 1 from pair i when Dr occurs, DrÎΩ.

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Proof: From definition of z, we know the possible values of z D is either zero or one. In constraints (iii), the 1i( r)

optimal solution must have z D = for only one i. 1i( r) 1

From (i), it is obvious that the optimal values of z D , 1i( r) Dr Î Ω, have direct relationship with the assignments

associated with pairs (λi, μi).

4. THE OPTIMAL SELECTION

The linear program (3), with constraints (4), (5), and (6) is the basis for the results we present in this section. To simplify the discussion, we first start to investigate the redundant constraints among (4). Since the state trajectory has to satisfy with (4), we have

0 1 ( ) ( ) 0. k i i j i j j j j x x w z w = +

å

³

Consider for i = 1 and any wk+1ÎW , (4) becomes k+1

1 1 1 0 1 1 1 ( ) ( ) ( ) 0 if , 1, 2,..., , k i j j k j j k j j k j x z z A A j m x w w w w + = + + + ³ = =

å

and 1 1 1 1 0 1 1 1 ( ) ( ) ( ) 0 if , 1, 2,..., . k j j k j j k j j k j x z z D D j m x w w w w + = + + - ³ = =

å

That is 1 1 1 1 1( k ) ( k) 1( k ) 0 k k k j x w + x w z w A + = + + ³ and 1 1 1 1 1( ) ( ) 1( ) 0 k k k k k k j x + w + =x w -z + w D ³ Hence, if 1 1 1 1 1( k ) 0 for all k k k x w + w + + + ³ Î W then it implies 1( k) 0 for all k k k x w ³ w Î W because of 1 1( ) 0 k k j z + w D ³

By induction from k = 1 to k = N - 1, we know constraint (4) with k = 1, 2, …, N - 2 is redundant if k =

N - 1 is satisfied by (4). Furthermore, among the

constraints (4) with k = N - 1, there are some other redundant constraints. Consider for any wNÎW , N

1 1 1 1 1 1 ( N) ( N ) ( N) ( N). N N N N x w x w - x w z w -= + If we have 1 1 1 1 1( N ) ( N ) 0 for any 1 , N N j x w - z w -D j m - - ³ £ £

then it is immediately satisfied with

1 1 1 1 1( ) ( ) 0 N N N N j x - w - +z w -D ³ because of 1 ( N 1 ) 0. N j z w -D ³ Therefore, instead of considering 1 ( N) 0 N x w ³ for every wNÎ WN , we consider only 1 1 1 1 1 1 0 1 ( ) ( ) ( ) 0 when , 1, 2,..., . N r r N r r N j j N j x z z D D j m x w w w w -= + - ³ = =

å

(7) for every wrÎ W , 1r £ £r N- ,1 i m1 £ £ .

In order to characterize the optimal solutions satisfying (4)-(7), its dual variables is introduced. Let dual variables y = (y1, y2, …, yN) in which y’s components yki(wk) 0³ , i =

1, 2, …, m + n - 1, k = 1, 2, …,N - 1 be defined associated with constraints (7). Rewrite the LP as

1 1 1 1 1 1 0 1 1 1 min ( ) ( ) ( )[ ( ) ( )] k k k k N m n i k i k k k k k N m n k i k i i k i k k k k k k j z y x z w w g w w w x w w - + -= ÎW = - + -= ÎW = = - +

å å å

å å å

å

subject to (4) and (6).

The cost function may be rewritten as

1 1 1 1 ( ) ( ) k k N m n i k i k k k k k c z C w w w - + -= ÎW = +

å å å

(8) where 1 1 1 ... ( ) {[ ( ,..., ) ( ) ] ( ) ( )} k j N i k k i k i j k j k k i j i j j j c Pr s y w w w w w x w b w x w + -= = +

-å -å

(9)

and C is a constant independent of z. In order to find the set of optimal solutions, we apply the complementary slackness theorem, namely, z is an optimal solution of (3)-(6) if and only if there exists y such that (a), (b) and (c) below hold.

(a) z is an optimal solution of (8) subjecting to (5) and (6).

(b) (Feasibility): i( k, ) 0

k

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(c) (Complementary slackness): If i( k) k y w >0, then ( , ) i k k

x w z = , for all i and k. 0

1 if ( ) 0 ( ) 0 if ( ) 0 [0,1] if ( ) 0 i k k i k i k k k i k k c z c c w w w w ì < ï =í > ïÎ = î

Clearly, (8) with constraints (6) has become an assignment problem with variables z and associated linear cost functions c depending on pairs of service and arrival rates as well as the switching cost. Because of

1 ... m 0

l ³ ³l > and 0<m1£ £... mm, if si = 0, we have

1

0 ... 0m

c £ £ . c

With constraints (6) and discussion above, the optimal solution is clearly decided from pair 1 to pair m one by one. In addition, based on sensitivity analysis of (8), every optimal solution of z has a correspondent interval associated with its coefficient c. When the initial queue size vector is large enough so that all constraints (7) satisfy with strict inequality, it implies y = 0 by complementary slackness conditions (c). In this case, c is not changed according to (9) and the optimal solution z remains the same for x0 > L, with some L > 0. It suffices to establish

the following lemma.

Lemma 4.1 The optimal solution will remain the same if the initial

queue size is greater than constants Ll, l = 1, 2, …, n, for every

queue l .

The proof is omitted because it is clear from the previous statement.

Lemma 4.2 The optimal solution will remain the same if the initial

queue size belongs to the interval [L Ll, l], l = 1, 2, …, n, for

every queue l .

Proof. Following Lemma 4.1, we only show 1

0

x shall be

greater than a constant L to satisfy the constraint (6). 1

Consider constraints (6), i.e., for every wN-1Î WN-1, 1 1 1 1 1 1 0 1 ( ) ( ) ( ) 0. N j j N j j N j x -x w z w z w -= +

å

- ³

Suppose wN =w w w1 2... N-1Di is given as a specific

transition path such that 1( j) 0 j

z w ¹ , for some j,

1£ £j N. Since the sum of some ( j 1 )

j i z w -D is equal to 1, we have 1 1 1 1 1 0 1 1 ( ) ( ) ( ). N j j j j N N i j x -x w z w z w -D = ³ -

å

-To restrict l only to denote i = 1, 2, …, m, we have showed that with an optimal policy an interval corresponds to each class i of a pair (λi, μi). Combining with the fact that

(8) is piecewise linear, it completes the proof.

Notice that these intervals may be ordered by a monotonic property of (λi, μi) but there maybe exists

overlaps between adjacent intervals, e.g., Li <Lj <Li,

and Lj <Li <Lj, for j = i + 1.

Theorem 4.1 The optimal policy for finite horizon in (1) has a

monotone hysteretic structure.

Proof. From Lemma 4.2, it suffices to show that the optimal policy will be switched from (λi, μi) to (λi+1, μi+1)

and switched back from (λi+1, μi+1) to (λi, μi) with different

bounds respectively. Since the coefficient of C D1i( r),

r

D Î W , is a function of λi or μi depending on the event at

the decision epoch, the optimal policy may be assigned when the initial queue size is increasing to Li+1 or

decreasing to L which depends on the initial queue size i

and the current service and arrival rates.

By induction on N, the results are easy to extend to a case of infinite horizon. We state the following theorem. Theorem 4.2 The optimal policy for infinite horizon in (2) has a

monotone hysteretic structure.

Proof. Applying Lippman (1975), Kumar and Meyn (1996), the result of finite horizon is able to extend to the case of infinite horizon.

5. DISCUSSION

The major contribution of this paper lies in the development of a new methodology for studying controlled queueing systems. We strongly believe that the LP based methodology has significantly higher potential than the traditional Dynamic Programming (DP) and Stochastic Dominance techniques. Since the proof is not possible to be provided by the DP approach, the strength of the approach stems from the following facts:

(a) It captures the essence of the dynamics and cost structure of the system (linear), without the burden of the (state-dependent) statistical descriptions;

(b) The sample path constraints of the system are conveniently ordered in the constraint matrix.

In general, the way the system parameters affect the parameters of the linear program are the key to the success of this approach. The parameters of the LP are very easily derived from the system parameters.

The second contribution of this paper is the derivation of the structure of the optimal policy in the important queueing problems, namely scheduling policies. With the study that has been done on deriving the necessary conditions of optimality and on reducing the original problem to a trivial assignment problem, the methodology presented in the paper is very

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general. One can use the approach to develop the structure of optimal policies for other abstract queueing network models as well.

6. CONCLUSIONS AND FURTHER RESEARCH We have shown that the optimal selection of arrival and service rates at the first queue of n queues in series has a monotone hysteretic structure. We applied the linear programming arguments to establish this result. If our assumption of αi for i = 1, 2, …, m is relaxed to consider more

general cases it is clear that 0

i

c may not have a monotonous

property when the queue length increases, i.e., the monotone hysteretic optimal policies may not exist. However, this should be studied for future research.

We believe that many optimal control queueing problems, in which the dynamic programming formulation fails, can be treated successfully via Linear Programming techniques. It is an effective analysis tool for obtaining performance structure in stochastical optimization problems.

REFERENCES

1. Bertsimas, D. and Niño-Mora, J. (1999). Optimization of multiclass queueing networks with changeover times via the achieveable region approach: part ii, the multi-station case. Mathematics of Operations Research, 24(2): 331-361.

2. Ghoneim, H. and Stidham, S. (1985). Optimal control of arrivals to two queues in series. European Journal of

Operations Research, 21: 399-409.

3. Hajeck, B. (1984). Optimal control of two interacting service stations. IEEE Transactions on Automatic Control, AC-29(6): 491-499.

4. Kumar, P.R. and Meyn, S.P. (1996). Duality and linear programs for stability and performance analysis of queueing networks and scheduling policies. IEEE

Transactions on Automatic Control, 41(1): 4-16.

5. Lippman, S. (1975). Applying a new device in the optimization of exponential queueing systems.

Operations Research, 23: 687-710.

6. Lu, F.V. and Serfozo, R.F. (1984). M/M/1 queueing decision processes with monotone hysteric optimal policies. Operations Research, 1117-1132.

7. Luh, H. and Rieder, U. (2001). Optimal control of arrivals in tandem queues of constant service time.

Mathematical Methods of Operations Research, 53(3):

481-491.

8. Luh, H. and Viniotis, I. (2002). Threshold control policies for heterogeneous server systems. Mathematical

Methods of Operations Research, 55(1) :121-142.

9. Moustafa, M.S. (1992). Optimal control of arrival and service rates for two M/M/1 queues in series.

Proceedings of the 1st World Congress of Nonlinear Analysis,

Tampa, Florida, USA, 19-26.

10. Puteramn, M. (1994). Markov Decision Processes: Discrete

Stochastic Dynamic Programming. Wiley, New York.

11. Rosberg, Z., Varaiya, P., and Walrand, J.C. (1982). Optimal control of service in tandem queues. IEEE

Transactions on Automatic Control, AC-27(3): 600-610.

12. Weber, R. and Stidham, S. (1987). Optimal control of service rates in networks of queues. Advances in Applied

數據

Figure 1. A series of  queues in tandem where  ( ,   ) l m i i   is selected at station 1

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