IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5, SEPTEMBER 1990 1173

bent sequences in all. The 448 16-bit sequences of weight 10 are the bitwise complements of these. For sequences of order 64 we calculate 37 879 808 BBBS’s and 10 321 920 LBBS’s, or 48 201 728 bent sequences in total. Although an exhaustive search through all 2@ vectors is cornputationally infeasible, it would be interesting to find out whether this is again a complete enumer- ation.

One direction for further research in this area is the search for a good method for calculating w in (2). Using the weight of a bent sequence and the requirement that E:=, y , = 1 (mod 2), we see that w is upper-bounded by 2 x ( p z : / 2 ) ; in fact, for y j of order four ( P Z k = 8) this bound is exact. We conjecture that w may be lower-bounded by

2x( P Z k / ’ ) = y ( ~ - l ) , P 2 k P 2 k for P Z k > 8 ,

but this is unproven.

REFERENCES 0. S . Rothaus, “On ‘bent’ functions,” J . Cornbin. Theory, vol. 20(A), pp. 300-305, 1976. R. Yarlagadda and J . E. Hershey, “Analysis and synthesis of bent sequences,” IEE Proc. ( f a f l E ) , vol. 136, Mar. 1989, pp. 112-123. J . D. Olsen, R. A. Scholtz, and L. R. Welch, “Bent-function sequences,” IEEE Trans. Inform. Theory, vol. IT-28, no. 6, pp. 858-864, 1982. A. Lempel and M. Cohn, “Maximal Families of Bent Sequences,” IEEE Trans. Inform. Theory, vol. IT-28, no. 6, pp. 865-868, 1982. P. V. Kumar and R. A. Scholtz, “Bounds on the linear span of bent sequences,” IEEE Trans. Inform. Theory, vol. IT-29, no. 6, pp. 854-862, 1983. F. J. MacWilliams and N. J. A. Sloane, The Theory ofEror-Corecfing Codes. New York: North-Holland, 1977. V. V. Losev, “Decoding of sequences of bent functions by means of a fast Hadamard transform,” Radiotechnika i elektronika, vol. 7, pp. 1479-1492, 1987. W. Meier and 0. Staffelbach, “Nonlinearity criteria for cryptographic functions,” in Advances in Cryptology: Proc. of EUROCRYPT ’89, to appear. R. McFarland, “Family of difference sets in noncyclic groups,” J. Combin. Theory, vol. 15(A), pp. 1-10, 1973. J . F. Dillon, Elementary Hadamard Difference Sets, Ph.D. thesis, Univ. of Maryland, College Park, 1974. P. V. Kumar, R. A. Scholtz, and L. R. Welch, “Generalized bent functions and their properties,” J . Combin. Theory, vol. 40(A), pp. . . 90-107, 1985. H. Chung and P. V. Kumar, “A new general construction for general- ized bent functions,” IEEE Trans. Inform. Theory, vol. IT-35, no. 1, pp. 206-209, Jan. 1989.

131 D. Elliott and K. R. Rao, Fast Transforms: Algorithms, Analyses, Appli- cations.

141 R. Yarlagadda and J . Hershey, “ A note on the eigenvectors of Hadamard matrices of order 2**n,” Linear Algebra and Applications, vol. 45, pp. 43-53, 1982.

121

New York: Academic Press, 1982.

Monotonicity Properties for the Stochastic Knapsack KEITH W. ROSS, MEMBER, IEEE, AND DAVID D. YAO

Abstract -A stochastic system is considered in equilibrium with N servers, no waiting room, and K classes of customers. A class-k cus-

Manuscript received February 24, 1988; revised March 1, 1990. K. W. Ross was supported in part by AT&T Grant 5-27628 and in part by NSF Grant NCR 8707620. D. D. Yao was supported in part by the NSF Grant ECS- 8658157 and in part by O N R contract N 00014-84K-0465.

K. W. Ross is with the Department of Systems, University of Pennsylvania, Philadelphia, PA 19104.

D. D. Yao was with the Division of Applied Sciences at Harvard Univer- sity, Cambridge, MA. He is now with the Department of Industrial Engineer- ing and Operations Research, Columbia University, New York, NY 10027.

IEEE Log Number 9036391.

tomer requires b, servers and releases them simultaneously af€er a random period of time. This multiclass blocking system is motivated by loss networks that support a variety of traffic types (e.g., voice, video, facsimile). The effect of increasing the state-dependent arrival rates and the number of servers on the throughputs and blocking probabilities are considered.

I. INTRODUCTION

We consider a stochastic system with N servers and K classes of customers. A class-k customer requires b, servers and re- leases them simultaneously after a random time with mean p;’. The class-dependent distributions for the holding times are arbitrary. Interarrival times for class-k customers are assumed to be exponential with parameter Ak(n,), where nk is the number of class-k customers currently in the system. If a class-k cus- tomer arrives to find fewer than b, servers available, then it is blocked and assumed lost. We shall refer to this system as a “stochastic knapsack,” due to its resemblance to the classical knapsack model of combinatorial optimization (e.g., see Dernardo [ 11).

This multiclass blocking system is motivated by circuit-switched networks that support a variety of traffic classes (e.g., voice, video, facsimile), each of which has a unique arrival rate, hold- ing-time distribution and bandwidth requirement. By modeling a trunk with N circuits as the N servers, and b, as the bandwidth requirement of class-k calls, then our stochastic knapsack corre- sponds to a trunk supporting teletraffic with multiple bandwidth requirements. Other applications include parallel processing, where jobs require a varying number of processors, and memory management, where jobs from different classes require different amounts of memory.

Permitting the arrivals to be state-dependent enables us to model several interesting applications. If

A,(nk) = A,, k = l;.. , K (1)

then the arrival streams become independent Poison processes. This case shall receive special attention and shall be referred to as the multibandwidth single-trunk model (in contrast to the tree network model described next). For this case, an efficient algo- rithm has been developed by Kaufman [3] and Roberts [91 to determine average blocking probabilities and throughput for the various classes. Another example is when the kth arrival stream is generated by a finite source with population size Nk, for k = 1,. . . , K . This arrival mechanism, referred to as Engset arrivals in teletraffic engineering (e.g., see Schwartz [ 13]), can be captured by setting

A,(n,) = ( N , - n,)A,, k = l; . . , K

State-dependent arrivals also enable us to model a class of circuit-switched networks. In particular, consider the multiband- width tree network, as depicted in Fig. 1. The tree network consists of a common trunk (employed by all classes) with N

Fig. 1. Multibandwidth tree networks.

0018-9448/90/0900-1173$01.00 0 1990 IEEE

1174 IEFF TRANSACTIONS O N INFORMATION THFORY, VOL. 36, N O . 5 , SEPTFMBFR 1990

circuits, preceded by K access trunks, where the kth such trunk has N, circuits. In this model, a class-k call (i.e., customer) arrives according to a Poisson process with rate hk and requires one circuit on the kth access trunk and b, circuits on the common trunk. This tree network can be captured with the state-dependent arrival rates

Mitra [7] investigated integral representations to determine blocking probabilities for the single-bandwidth version of the above network (i.e., b, = b , = . . . = bK) . Tsang and Ross [17] developed efficient algorithms to determine exact blocking prob- abilities for multibandwidth tree networks, as well as more general hierarchical tree networks. Whitt 1191 and Kelly [5] studied the Erlang fixed-point equations to approximate block- ing probabilities for a much more general class of circuit-switched networks.

Note that the stochastic knapsack can also be viewed as a generalization of Erlang’s loss system (which corresponds to the case K = 1 and A,(n)= A). It is well known, and intuitively obvious, that the blocking probability and throughput are mono- tonically increasing functions of the arrival rate. Now consider the stochastic knapsack model with A,(nk) = A , , k = 1;. ., K (i.e., the multibandwidth single-trunk case). It is no longer entirely obvious whether the blocking probability of class-k customers will increase monotonically as A, increases. Indeed, increasing A, may block “wide” customers (i.e., customers from those classes with high bandwidth requirements), and open up circuits for future class-k customers.

In this correspondence, we study the monotonicity properties of performance measures for the stochastic knapsack in equilib- rium. We consider increasing the state-dependent intensities pk(n), where p,(n) := h k ( n ) / P k , and increasing N, the number of servers. In the case of tree networks, we also consider increasing N,, the number of circuits in the kth access trunk. Two assumptions are needed for many (but not all) of the results: the monotonicity condition, where Ak(n) is assumed to be nonincreasing in n for all k = 1 , . . . , K ; and the dicisibility condi- tion, where b k + , is assumed to be an integer multiple of bk for all k = 1 , . . ., K - 1. In particular it is shown under the previous two conditions that the knapsack utilization is increasing, in the sense of the likelihood ratio ordering, as p K ( . ) increases compo- nentwise. This result will aid us in demonstrating for the multi- bandwidth tree networks that the blocking probability of class-k calls will increase as a function of p, ( := A, / p , ) if either j = K or k = K. Furthermore, we shall show that this blocking proba- bility may be neither monotonically increasing nor monotoni- cally decreasing if j # K and k f K .

These results are obtained with techniques similar to those in Shantikumar and Yao [ 141, where monotonicity properties for closed queueing networks were studied. In particular, we com- bine the product-form solution for the stochastic knapsack with the notion of “equilibrium rate.” However, in contrast with the model in [14], the underlying random variables in the knapsack model have “gaps” in their support, and classical results on the the likelihood ratio ordering and on the convolution of PF, (Polya frequency of order 2; see Section 11) random variables are no longer applicable. It has therefore been necessary to develop a convolution and likelihood ratio theory for random variables with gaps.

For single-bandwidth tree networks, significantly stronger re- sults are obtained. For example, we show that the throughput for each class is increasing as the number of common circuits N

is increased. Moreover, throughput for class-k calls is increasing (resp. decreasing) as the number of circuits in the j t h access link N, increases with j = k (resp. j # k).

In related research, Ross and Tsang [ 111 considered accepting and rejecting customers, as a function of the number of cus- tomers of each class in the knapsack, to maximize average revenue. Techniques from stochastic ordering were employed to prove the optimality of threshold policies over the class of coordinate-convex policies (see also Ross and Tsang [ l o ] for a Markov decision approach, where optimization is carried out over all policies). Smith and Whitt [16] employ methods from stochastic ordering to prove that blocking systems can be made more efficient by combining two such systems. Also, Nain [8] considers monotonicity properties for the stochastic knapsack with K = 2, with emphasis on transient behavior.

This correspondence is organized as follows. In Section 11, a likelihood ratio and convolution theory for random variables with gaps is developed. In Section 111, the effect of increasing the arrival rates is considered. In Section IV, we consider the effect of increasing the knapsack size N and, in the case of tree networks, the case of increasing the capacity of the access trunk N k . In Section V, we investigate the relationship between single-bandwidth tree networks and closed queueing networks. We conclude in Section VI with a brief discussion on mono- tonicity properties for knapsacks with queueing.

11. PRELIMINARIES

For the purpose of this correspondence, we shall focus on random variables that take values in the nonnegative integers.

Definition 1: Let X be a discrete random variable such that P [ X = n] > 0 for n = 0 , 1 , 2 , . . . . The equilibrium rate of X , denoted as r , ( . ) , is a real-valued function defined as follows:

r ,dO) = 0; r x ( n ) = P[ x = n - I ] / P [ x = n ] , n = 1 , 2 , . . . .

Based on equilibrium rate, the PF, property (Polya frequency of order 2; see Karlin and Proschan [ 2 ] ) can be expressed as

X E PF, - rx ( n ) increasing in n . (3)

(Throughout this section we use “increasing” and “decreasing” in the nonstrict sense.) The likelihood ratio ordering, abbrevi- ated 2‘‘ (see S. Ross [12]), can be expressed in terms of the equilibrium rate. If X and Y are two random variables with support on (0 ,1 ,2 , . . . ), then

(4)

It is well known that X 2“ Y + X 2’‘ Y - E [ f ( X ) I E [ f ( Y ) I for all nondecreasing functions f ( . ).

Throughout this section, let a be a fixed positive integer. Based on the PF, property of Definition 1, we can define the following “PF,(a)” property.

Definition 2: The random variable X is said to have the “PF,(a)” property, denoted X E PF2(a), if rx (ma + b ) is in- creasing in m for all b = 0,. . . , a - 1.

It is easily seen that X E PF,(a) if and only if r,(n) I r,(n + a ) for all n = 1,2, . . . . We will find it more convenient, however, to refer to Definition 2 in the subsequent proofs. Obviously, X E PF,(l) X E PF,. Also, it is evident from this definition that PF,(a) - PF,(na) for any positive integer n.

A classical result due to Karlin and Proschan [2 ] states that the PF, property is preserved under convolution; i.e., for two

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5, SEPTEMBER 1990 1175

independent random variables X and Y , if X E PF, and Y E

PF,, then X + Y E PF,. However, it is not difficult to verify that for discrete random variables, X E PF, and Y E PF, does not in general imply X + aY E PF,. The positive result is as follows.

Theorem 1: Suppose X and Y are two independent random variables with support on (0,1,2, . . . }. Then, X E PF,(a) and Y E PF, implies X + aY E PF,(a).

Proof Let p,, := P [ X = n ] and q, := P[Y = n ] . Let Z .= X + ay. Then

m

P i a + b - l ( l m - r

rz(ma + b ) = ( 5 ) Pia+bqm-r

r = O

It suffices to show rz(ma + b ) I rz((m + l)a + b). Note that m

Pia+h-lqm+l-r + P ( m + l ) a + b - I q O

r z ( ( m + 1). + b ) = '=Om

Pia + b q m + 1 - - I + P(m+ I ) a + b q O i = o

The desired inequality is then equivalent to m m

PJa+hqm+l- j + p ( m + l ) o + h q O

m

When a = 1, Theorem 2 recovers the known result cited earlier. Also note that the condition X E PF,(a) in Theorems 1 and 2 can be changed to X E PF,(d), with d being any divisor of a , including d = 1 and d = a.

Combining Theorem 1 and Theorem 2 gives the next theorem.

Theorem 3: Let (Yk, k = 1;. ., K } be a set of independent PF, random variables. Let ( b k , k = 1; . ., K } be a set of positive integers such that bk is a divisor of bk + for all k = 1 , . . . , K - 1. Then Y, 2" Y i implies blYl + . . . + bKYK 2"blY, + . .. + b, Y; .

Notice that the conclusion in Theorem 3 applies to Y,, but does not apply to the other random variables Yk, k # K.

Two points are worth noting before we conclude this section.

1) Suppose / = { i o , i 1 ; ~ ~ , i , } MI^) is the support of X , where io < i, < . . . < i, are integers. Then, define

For two random variables X and Y with the same support 9 as before, X 2" Y is still meaningfully defined as r X ( n ) s r y ( n ) , for all n = 1; . ., M . In fact, we can even allow Y to have a "smaller" support set of the form / ' = { i O , z l ; . . , i M t } with M' < M . In this case, we simply define r , ( n ) = m, for all n > M'. Moreover, it is not difficult to show that X 2 " Y continues to imply X 2" Y with this extended definition of likelihood ratio ordering (a fact that will be of use in the subsequent section).

2) Suppose in Theorem 1 X has support on (0,l; . ., N } and Y has support on (0,l; . . , M } , with 0 < M < N (this will occur in tree networks). It can be verified that the proof of Theorem 1 is still valid for this case. The proof of Theorem 2 also remains valid when X , Y , and Y' have different support sets, with the support set of Y being larger than that of Y . Theorems 1 and 2 also remain valid when X (instead of Y ) has a larger support set.

r X ( 0 ) = 0; r X ( n ) = P [ x = i n - , ] / P [ x = i n ] .

111. INCREASING TRAFFIC INTENSITIES

Let R be the set of states for the stochastic knapsack, i.e., This inequality holds, since 1) the two parentheses under the = {n: nb I where . and = ( b , , . . . ,bK),

PF2(a), Y E PF, and i > j ; and 2 ) the parenthesis in the second < , . , - 5 b,. Denote X =(XI , . . ., X,), where xk is the equilib- rium number of class-k customers in the system. It is well known summation is 2 0, due to X E PF,(a).

first respectively9 and '7 due to E Without loss of generality, we assume throughout that b , b ,

It is obvious that when a = 1, Theorem 2 recovers the classical result of Karlin and Proschan.

A known result regarding likelihood ratio ordering is as follows (see, e.g., Keilson and Sumita [4]): if X E PF, is inde- pendent of Y and Y', then Y yrr Y' implies X + Y 2 " X + Y'. We shall need the following generalization of this result.

Theorem 2: Suppose X E PF,(a) is independent of Y and Y', where X , Y, and Y have support on (0, 1,2, . . .]. Then, Y krr Y' implies x + aY 2"X + aY'.

Proo$ Following the notations in the proof of Theorem 1, we have rz(rna + b ) expressed in (5) . Denote 2' := X + aY and q; := P [ Y = n]; then r z h a + b ) is obtained by replacing q in (5) with 4 ' . We want to show that rz(ma + b ) I r,.(ma + b). Similar to the proof of Theorem 1, this is equivalent to

(Pia+b-IPJa+b - P i a + b ~ J a + b - 1 ) ( ~ m - l ~ h - J - q m - j q h - I )

1 > J

I O . This inequality holds, since the first parentheses is 2 0, due to X E PF,(a), and the second parentheses is 5 0, due to Y >'r Y'.

0 where Z := by.

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1176 I E E E TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5 , S E P T E M B E R 1990

Now let Yo be a random variable independent of (Y,, k =

1; . ., K ) with probability mass function

where

c = L + l / ( l - c u ) , O < a < l and L 2 N .

Note that Yo E PF,. For a given random variable W , we shall denote W := W + Yo. Then (7) can be written as

K n p [ y k = n k l

p [ X = ” ] = k = o (8) P [ i = N ]

where no = N - bn. Before beginning our study of monotonicity, it is convenient

to introduce some additional notation. Let X:= (1,. . . , K } and for any subset J G X denote

B,,:= b k X k , B : = B , , B o , : = B - b , X , , k e J

z,:= bkYk and Z ( , ) = z - bkYk. k e g

Note that B is the equilibrium number of servers busy, which reflects the utilization of the knapsack. Further note that the throughput of class-k customers, denoted TH,, is given by

(9)

Denote P k = (pk(o) , pk(1); ’ ‘, p k ( N ) ) and = ( h k ( O ) , A,(l); . ., A k ( N ) ) . In what follows, the statement “ P k increases” should be interpreted as “p, increases componentwise.”

Proposition I : For k = 1; . . , K , xk is increasing with respect to pk in the likelihood ratio ordering. Consequently, TH, is increasing in A,, k = 1;. ., K.

Proof: From (8) we have

P [ Yk = n ] P [ g ( k ) = N - bkn] P [ x, = n ] = P[i = N ]

from which we have

Since ry$n) = n / p k ( n ) and since the denominator of (10) does not involve p k , the first statement follows. The second statement

0

For the remainder of this correspondence, we shall assume

Monotonicity Condition: For all k = 1,. . . , K , p , (n ) is nonin-

Divisibility Condition: For k = 1,. . . , K - 1, b, is a divisor of

Note that the monotonicity condition implies that Yk E PF,, for

We briefly note that the divisibility assumption can be ex- pected to be satisfied in practice. For example, in the multiplex- ing hierarchy in North America, the most important basic chan-

follows from the first along with (9).

that the following assumptions hold true.

creasing in n.

b k + l *

9 K. k 1, . . .

nel rates for telephony are DSO, DS1 or T1, and DS3. These rates correspond to b , = 1, b , = 24, and 6 , = 28 X 24.

Henceforth, when we increase A, or p k , we shall assume that they are increased in such a way that the monotonicity condition remains satisfied. These two conditions enable us to strengthen the result of Proposition 1 as follows.

Proposition 2: Let 9 be a nonempty subset of X. Denote 2? = X - 9 and k for the largest element in 9. Then B,g is increasing and B, is decreasing in P k in the likelihood ratio ordering. In particular, the knapsack utilization B is increasing in pK in the likelihood ratio ordering.

Proof: Let i , , , i , ; . . , i M be the support of B,g, where i n < i n + , , n = 0;. ., M - 1. From (8) it follows that

P [ Z , = i , l ~ [ .2x = N - i n ] P [ B , = i n ] =

P[i= N ]

from which it follows that

. .“ rn = i n - ,

Now increasing P k will increase Yk in the likelihood ratio ordering. Thus, by Theorem 3, increa$ng p k will increase Z, in the likelihood ratio ordering. Since Z , does not involve pk, it follows that B, is increasing in pk in the likelihood ratio ordering. The fact that B, is decreasing is similarly proved by interchanging 9 and A? in (ll), and defining (io,. * , i M } to be

If we let 2 = ( k } in Proposition 2, and take (9) into account, we obtain the next corollary.

Corollary 1: For k = 1,‘ . . , K - 1, X , is decreasing in pK in the likelihood ratio ordering; consequently, for k = 1; . ., K - 1, TH, is decreasing in A,. Moreover, X , is deceasing in p K - , in the likelihood ratio ordering; consequently, TH, is decreasing in A,- ,.

It is important to note that Proposition 2 (and hence Corol- lary 1) may not hold true when the divisibility assumption is not satisfied. To see this, consider the single-trunk model with N = 4 , b , = 1, b , = 2, and b , = 3. We claim that TH, can actu- ally increase with A,, contradicting Corollary 1. The argument goes roughly as follows (it can be made rigorous; see that proof of Proposition 4). Suppose that pk >> 1, k = 1,2,3, so that P ( B = 4 ) = 1. Further suppose p, >> rnax(p,,p,), so that P ( X , =0, X , = 2 , X,=O)=l. In this case TH,=p,E[X,]=O. Now increase A , so that p , becomes s p , . We will then have P ( X , = 1, X , = 0, X , = 1) = 1 and TH, = p , E [ X , ] = p, , estab- lishing the claim.

Remark: In Proposition 2, it is only necessary that the divisi- bility condition hold over the subset 9. Therefore, the divisibil- ity requirement can be relaxed also in Corollary 1. As an example, consider the case K = 3. From Proposition 1, we know that TH, is increasing in A,, k = 1 ,2 ,3 (without a divisibility requirement). From Proposition 2 (with the relaxed divisibility requirement), we know that: 1) if b , / b , is an integer, then TH, is decreasing in A,; 2) if b , / b , is an integer, then TH, is decreasing in A,; 3) if b , / b , is an integer, then TH, is decreasing in A,. Similarly, for the case K = 2 we know that without any divisibility requirement the following holds true: TH, is increasing with A, and decreasing with A, ; TH, is decreasing with A,.

the support of B,. 0

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5 , SEPTEMBER L990 1177

A. Multibandwidth Tree Networks We now suppose that the arrival rates are given by (2).

Equivalently, the arrival streams can be viewed as K indepen- dent Poisson arrival processes with rates A,, k = 1; . ., K , where class-k calls are blocked if either B > N - 6, or X k = Nk. We therefore have the following formulas for blocking probability:

(13)

where (7) has been employed to obtain the last equality. Note from (12) that, for j # k, P k is increasing in p j if and only if TH, is decreasing in hi.

Proposition 3: p k is increasing in pi, 1 I j , k I K , if either j = K or k = K.

Proofi From Corollary 1 and (12) we know for all k =

We now make use of an elasticity result due to Virtamo [18], 1; . ., K - 1, that p k is increasing in p,.

which states that for all 1 I j , k I K

apj - a p k

aPk ' P j

for the system under consideration. In particular, we have dp, / d p k = dp, / d p K , which, when combined with the result of the previous paragraph, gives that PK is increasing in p k ,

It remains to show that pK is increasing in p,. Consider (13) with k = K. Note that Y, is increasing in p K in the likelihood ratio ordering, as well as Z and B , due to Theorem 3 and Proposition 2, respectively. Since likelihood ratio ordering im- plies stochastic ordering, it therefore follows that P[Y, = N,] = P[Y, 2 N,], 1 / P [ Z I N ] and P [ B > N - b K ] are all increasing in p,. Also note that Z(,) does not involve p K . Combining these

0

The previous theorem has an obvious analog in terms of the various throughputs. Indeed, from Proposition 1 , Proposition 3 and (12), the Jacobian matrix for the throughputs has the following form:

k = l ; . ., K - 1 .

facts with (13) for k = K gives the desired result.

- + + - I1 +

[T] = [ + + Is j ,ksK - - - -

where the plus sign and minus sign correspond to positive and negative derivatives, respectively. Note that the signs of many of the derivatives are not accounted for in the previous Jacobian matrix. Indeed, TH, is not necessarily monotonically increasing nor decreasing in Ai, if j # k, j # K and k # K (i.e., a blank in the previous Jacobian matrix corresponds to a derivative that can be either positive or negative.) We establish this fact for the special case of arrivals of the form ( 1 ) (equivalently, Nk 2 N , k = 1; . ., K ) .

Proposition 4: Consider the multibandwidth single-trunk model. Fix j , k such that 1 I j , k I K - 1. Suppose that b, is a divisor of N and that b, 2 2b,-,. Then there exists p+ =

( p ; ; . . , p , ' ) and p-=(p;; . . ,pK) , with p+>O, p,:>O, 1 1 i - < K , such that

' p k ap k - (p ' )>O, - ( p - ) < O . ' p j ap,

Proot From the product-form solution (6) , we have

G ( N - b k )

G ( N ) f i k = 1 -

where G ( N ) can now be written as

From (14) it is easily seen that 0 < p k < 1 and, due to bj being a divisor of N (since bj is a divisor of b, 1, that p k + 1 as pi + m.

Hence, there exists a p + at which dp, / d p j is strictly positive. From (14) it also follows that d p k / d p j is strictly negative if

and only if

a G ( N ) > 0. (15) dG( N - b k )

- G ( N - b k ) - ' P j

Let fi = (0,. . . ,o, P,), where p K > 0. For any integer F we have

and

F - bj

where 1x1 is the largest integer less than or equal to x and

Thus, (15) holds true at fi if and only if

Since b, 2 2b,-, and b,, b, I b,-,, we have

N - b k - b , N - b k N - 6 , [ b, ]=[Tl=[b,l=a-' where a := N/b,. Therefore, (16) is equivalent to $(a )> 4(a - l) , which clearly holds true. Hence, (15) holds true at fi.

To complete the proof we note that G ( F ) and dG(F)/dp, are continuous functions of p over [O,mIK (this fact follows immedi- ately from the definition of G(F)) . Hence, (15) holds true for

0 some p- and p,- > 0, i = 1; . ', K.

B. Single-Bandwidth Knapsacks In this subsection, we assume that b , = b, = . . . = b, = 1.

Thus, the classes can be ordered in any desirable manner and the divisibility condition will be trivially satisfied. As a conse- quence of Proposition 2 and (9) we have the following result.

Corollary 2: Let 9 be a nonempty subset of X and let k E 9. Denote 2? = X - d. Then B , is increasing and B , is decreasing in p k in the likelihood ratio ordering. Consequently, THk is decreasing in A, for j # k.

1178

Due to the previous result and Proposition 1, the Jacobian matrix for the throughputs has the following form for single- bandwidth systems:

Note that significantly more can be said about the monotonicity of throughputs for the single-bandwidth case. As a consequence of Proposition 3 we have the following results for single-band- width tree networks.

Corollary 3: Suppose that the arrival rates take the form (2). Then Pk is increasing in pi, for all 1 I j , k I K .

IV. INCREASING CAPACITY

We first consider the problem of increasing the number of servers N for the general knapsack model. The following result requires neither the monotonicity nor the divisibility condition.

Proposition 5: E is increasing in the likelihood ratio ordering as N increases.

Proofi For a system with N servers, let {iI;..,iM) be the support of E. Let E' denote the utilization for a system with N + 1 servers, and let ( i l , * . . , iM,} be the corresponding support, where M 'z M. Similar to the derivation of (111, we have

r B ( n ) = r z ( n ) , n = l ; . . , M

and

rB,( n ) = r z ( n ) , n = 1; . . , M'.

Thus, rB.(n) I rB(n) , n = 1; . ., M', which completes the proof. 0

With the monotonicity and the divisibility conditions, we can make the following conclusion.

Proposition 6: Let 9 be such that 0c 9 c X, Denote X = X - d and k for the largest element in 2'. Then, increasing N by b, will increase E, f in the likelihood ratio ordering.

Pro05 For the system with N servers, let {io, i,, * . . , iM} be the support of E,9. For the system with N + b, servers, desig- nate all parameters with a prime (i.e., M', E,>, etc.). Also, let L appearing in the definition of Yo satisfy L 2 N + b,.

From (11) we have

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5 , SEPTEMBER 1990

and

Therefore B , <"E> if

r i , (N+ b, - m) z r2r (N- m), m = O ; . . , N. (17)

But from Theorem 3, 2, E PF,(b,), which implies (17). U

Note that when 3 = X, Proposition 6 can be strengthened to Proposition S .

Letting #={k} in Proposition 6 and utilizing (9) gives the following result.

Corollary 4: If N is increased by b,, then for k = 1; . . , K - 1, TH, is increased. If N is increased by b,- then TH, is increased.

A. Multibandwidth Tree Networks In this subsection, we assume that the arrival rates follow (2).

From Corollary 3 and (12) we have the following result. Corollary 5: If N is increased by b,, then for k = 1; . ., K - 1,

0, is decreased. If N is increased by b,-,, then PK is de- creased.

Now consider the problem of increasing N,, the capacity of the kth access trunk. This increases A, (see (2)), so that Propo- sitions 1 and 2 apply. For example, we have Corollary 6.

Corollary 6: p,, k = 1 , . . ', K - 1 is increasing N,.

E. Single-Bandwidth Tree Networks

In this subsection we suppose that the arrival rates have the form (2) and that b, = b, = . . . = b,. From Corollary 2 and Corollary 5 we have (for example) the next result.

Corollary 7: Increasing Ni will increase Xi and decrease xk, k # j , in the likelihood ratio ordering. Consequently, increasing NI will increase THi and decrease TH,, k # j . Increasing N will increase X, , k = 1; . ., K, in the likelihood ratio ordering. Con- sequently, increasing N will increase TH,, k = 1; . . , K . An analogous result holds for blocking probabilities.

For single-bandwidth tree networks we can show that TH, is not only increasing, but is also concave in N. To see this, recall that ry$n) = n / p k for n = 1; . ., N,. Define r , (n) = Nk / p k for N > N,.AThen r,$n> is an increasing concave function of n. Since Z = Yo + Y, + . . . + Y,, it therefore follows from Shantikumar and Yao [lS] that r i ( n ) is increasing and concave in n. Now let ak be the probability of acceptance of a class-k call (i.e., a, = 1 - 0,). Then, for al k = 1; . ., K,

a, = P ( E I N - 1) = P ( 2 = N - 1 )

P(i = N )

= r i ( N ) .

Hence, a, is concave in N, for k = 1; . ., K . This is turn implies that TH, and P k are concave and convex, respectively, in N for all k = 1;. ., K.

V. AN EQUIVALENT CLOSED QUEUEING NETWORK FOR THE SINGLE-BANDWIDTH TREE NETWORK

Consider a closed queueing network consisting of a central node which feeds K auxiliary nodes (nodes 1 through K ) , as shown in Fig. 2. Suppose N customers circulate in the network. Let n =(n,;. . ,n , ) be the state of the network, where nk is the number of customers present at node k. (Thus, the number of

_j N jobs

~_______ ~ ~

Fig. 2. Equivalent closed queueing network for single-bandwidth tree network.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 36, NO. 5, SEPTEMBER 1990 1179

customers present at node 0 is N - n , - . . . - n, . ) Let the state-dependent service rate at node k, k = 1;. ., K, be n k p k . Let the state dependent service rate at the central node be A := A , + . . . + A, if n , + . + nK < N and 0, otherwise. Let the routing probability from the central node to node k, k = 1,. . . , K , be hk / A . Initially assume that the buffer capacity is infinite at each of the nodes. Note that this closed queueing network is a migration process with reversible routing (see p. 42 of Kelly [6]). Thus, the (vector) sate process for this network is reversible.

Recall that a single-bandwidth tree network consists of a common link with N circuits and access links with Nk circuits, k = 1;. ., K . If Nk 2 N for all k = 1;. ., K, then the previous closed queueing network would clearly be equivalent to our single-bandwidth tree network.

Now consider the previous queueing network with finite buffers N k , k = 1; . ., K , at the auxiliary nodes. Suppose that blocking operates as follows. When a customer is routed to a node k, k = 1; . ., K , whose buffer is full (i.e., nk = Nk) , the customer is instantaneously routed back to the central node and the central server continues to work. It is apparent that this closed queueing network with blocking is equivalent to our single-bandwidth tree network. It is interesting to note that the closed queueing network with blocking remains reversible, since it corresponds to a truncated reversible process [6].

It is also interesting to note that the single-bandwidth tree network can be regarded as the limit of a sequence of nonblock- ing closed queueing networks. Indeed, consider again the previ- ous closed network where each node has infinite buffer capacity. Now let the service rate at node k, k = 1; . ., K, be equal to n k p k if nk I Nk and equal to 1 / ~ if nk > Nk. This network is a standard nonblocking closed queueing network, for which there is a well-known product-form solution. It is straightforward to show that this product-form solution approaches that of the single-bandwidth tree network as E LO.

VI. CONCLUSION We conclude by mentioning that the theory developed in this

paper can be extended to the case where queueing is permitted in the knapsack. Indeed, we can allow the service rates to be state-dependent (in addition to state-dependent arrival rates), taking the form pk(n), n = 1;. . , N . In this case, we define pk(n) = A k ( n ) / p k ( n + l), and interpret to the monotonicity con- dition accordingly.

With state-dependent service rates, the throughput of class-k customers becomes

THk = E [ p k ( X k ) l .

Assume that p & ( n ) is nondecreasing in n , so that if xk in- creases in the likelihood ratio ordering, then so does THk. With this added condition, Proposition 1, Proposition 2, and Corollary 1 continue to hold true (with no change in their proofs). Note that in the case p & ( n ) = pk for n = 1; . ., N , then the queueing system has K servers, with one server dedicated to each cus- tomer class.

ACKNOWLEDGMENT Useful discussions with Philippe Nain of INRIA during the

early stages of this research are gratefully acknowledged.

REFERENCES [ l ] E. V. Denardo, Dynamic Programming: Models and Applications. En-

glewood Cliffs, NJ: Prentice-Hall, 1982. [2] S. Karlin and F. Proschan, “Polya-type distributions of convolutions,”

Annals of Mafh. Sfafisf., vol. 31, pp. 721-736, 1960. [3] J. S. Kaufman, “Blocking in a shared resource environment,” IEEE

Trans. Comm., vol. Com-29, vol. IO, pp. 1474-1481, 1981.

J. Keilson and U. Sumita, “Uniform stochastic ordering and related inequalities,” Canadian J. Sfafisf., vol. IO, pp. 181-198, 1982. F. Kelly, “Blocking probabilities in large circuit-switched networks,” Advances Appl. Probability, vol. 18, pp. 473-505, 1986. F. P. Kelly, Reiwsibility and Sfochastic Networks. Chichester, England: Wiley, 1979. D. Mitra, “Asymptotic analysis and computational methods for a class of simple, circuit-switched networks with blocking,” Adu. Appl. Prob., vol. 19, pp. 219-239, 1987. P. Nain, Qualitative properties of the Erlang blocking model with heterogeneous user requirements,” to appear in Queueing Sysfems: Theory and Applications. J. W. Roberts, A service system with heterogeneous user requirements,” Performance of data communicafions systems and fheir applications, pp. 423-431, 1981. K. W. Ross and D. Tsang, “Optimal circuit access policies in an ISDN environment: A Markov decision approach,” IEEE Trans. Commun., vol. 37, pp. 934-939, 1989. -, “The stochastic knapsack problem,” IEEE Trans. Commun., vol. 37, pp. 740-747, 1989. S. Ross, Stochastic Processes. M. Schwartz, Telecommunicafion Networks: Protocols, Modeling and Analysis. Reading, MA: Addison-Wesley, 1987. J. G. Shanthikumar and D. D. Yao, “The effect of increasing service rates in a closed queueing network,” J. Appl. Prob., vol. 23, pp. 474-483, 1986. -, “Second-order properties of the throughput of a closed queue- ing network,” Mafh. Operations Res., vol. 13, pp. 524-534, 1988. D. R. Smith and W. Whitt, “Resource sharing for efficiency in traffic systems,” Bell Syst. Tech. J . , vol. 60, no. 1, pp. 39-55, 1981. K. W. Ross and D. Tsang, “Teletraffic engineering for product-form circuit-switched networks,” to appear in Advances in Applied Probabil- ity. J. P. Virtamo, “Reciprocity of blocking probabilities in multiservice loss systems,” IEEE Trans. Commun., vol. 36, pp. 1174-1175, 1988. W. Whitt, “Blocking when service is required from several facilities simultaneously,” AT&T Tech. J., vol. 64, pp. 1807-1856, 1985.

New York: John Wiley, 1983.

Passage Times of Gaussian Noise Crossing a Time-Varying Boundary

A. J. SENIOR MEMBER, IEEE

Abstract -The first and second passage times of a stationary Gauss- ian process crossing a time-varying boundary were studied in a recent paper. The results apply to stationary Gaussian processes having a finite expected rate of level crossings, but they are restricted to time- varying boundaries having zero slope at the origin. This note extends the earlier results to include general time-varying boundaries. Some exact asymptotic probability densities of the first passage time of a stationary Gaussian process crossing a ramp or linear boundaries are developed. Also, some approximate results concerning the initial behaviors of the probability densities of the first passage times associated with a linear boundary are presented.

I. INTRODUCTION The special issue [l] in memory of S. 0. Rice contains a paper

by this author that treats the first and second times of passage of stationary Gaussian noise through a time-varying boundary. For simplicity, the time varying boundary -- S ( t ) was restricted there to have zero initial slope, i.e., S’(0) = 0. However, the more general case S’(0) # 0 is also of considerable interest since it includes the important case of a linear boundary. Surprisingly, this seemingly minor generalization of the initial slope of the boundary proved to be much more complicated than anticipated and could not be completed in time for inclusion in refer- ence [I].

Accordingly, for given initial conditions on the Gaussian pro- cess, this note extends the earlier results in [l] to include a general time-varying boundary.

Manuscript received March 28, 1989; revised December 10, 1989. The author is with AT&T Bell Laboratories, Whippany Road, Whippany,

IEEE Log Number 9036001. NJ 07981.

0018-9448/90/0900-1179$01.00 01990 IEEE