Cello, Marco

Gnecco, Giorgio

Marchese, Mario

Sanguineti, Marcello

English

CELLO, MARCO

GNECCO, GIORGIO STEFANO

MARCHESE, MARIO

SANGUINETI, MARCELLO

Archivio istituzionale della ricerca - Università di Genova

A Generalized Stochastic Knapsack Problem withApplication in Call Admission ControlMarco Cello, Giorgio Gnecco, Mario Marchese,Marcello SanguinetiDepartment of Communications, Computer and Systems Science (DIST), University ofGenoa - Via Opera Pia, 13 - 16145 Genova (Italy){marco.cello,giorgio.gnecco,mario.marchese,marcello}@dist.unige.itKey words: Stochastic Knapsack, Feasibility Region, Call Admission Control,Coordinate-Convex Policies.1 IntroductionIn the classical knapsack problem, a knapsack of capacity C is given, together withK classes of objects. For every k = 1, . . . , K, each object of the class k has a sizebk and an associated reward rk. The objects can be placed into the knapsack as longas the sum of their sizes does not exceed the capacity C. The problem consists inplacing the objects inside the knapsack so as to maximize the total reward.Among the various extensions to a stochastic framework available in the literature(see, e.g., [7,5,3]), we consider the stochastic knapsack problem proposed in [7].In such a model, objects belonging to each class become available randomly, ac-cording to exponentially-distributed inter-arrivals times with means depending onthe class and on the state of the knapsack. Each object has a sojourn time indepen-dent from the sojourn times of the other objects and described by a class-dependentdistribution. If put into the knapsack, an object from class k generates revenue ata positive rate rk. Let nk ≥ 0 denote the number of objects of class k that arecurrently inside the knapsack. Then one has the linear constraint∑k∈Knkbk ≤ C . (1)The problem consists in finding a policy that maximizes the average revenue, byaccepting or rejecting the arriving objects in dependence of the current state of theknapsack.The stochastic knapsack problem that we have just described has application, e.g.,in Call Admission Control (CAC) for telecommunication networks. In such a con-text, the objects are requests of connections coming from K different classes ofCTW2011, Villa Mondragone, Frascati, Italy. June 14–16, 2011users, each with a bandwidth requirement bk, k = 1, . . . , K, and a distribution forits duration. In CAC problems, often the constraint (1) arises as a linearization ofthe nonlinear constraint∑k∈Kβk(nk) ≤ C , (2)where the βk(·) are nonlinear nonnegative functions. The model in which the linearconstraint (1) is replaced by the nonlinear one (2) is known in the literature as thegeneralized stochastic knapsack problem 1 [4]. In call admission control, the coef-ficients bk of the linearized constraint are called effective bandwidths [6, Chapter1].The setsΩFR := {(n1, . . . , nK) ∈ NK0 :∑k∈Knkbk ≤ C}, (3)in the linear case, andΩFR := {(n1, . . . , nK) ∈ NK0 :∑k∈Kβk(nk) ≤ C} (4)in the nonlinear case, are called feasibility regions. In the context of admissioncontrol they model subsets of the call space {(n1, . . . , nK) ∈ NK0 }, where givenQuality of Service (QoS) constraints are satisfied.In general, finding optimal policies is a difficult combinatorial optimization taskboth for the stochastic knapsack problem [6, Chapter 4] and for the generalizedstochastic one [1,2]. The a-priori knowledge of structural properties of the (un-known) optimal policies is useful to find the solutions or, at least, good subop-timal policies. For two classes of objects and the linear constraint (1), structuralproperties were derived in [7] for the optimal policies belonging to the familyof coordinate-convex policies. Such properties restrict the K-tuple (n1, . . . , nK)to suitable subsets of the feasibility region ΩFR. Some extensions to nonlinearly-constrained feasibility regions of the structural results obtained in [7] for linearly-constrained ones were derived in [2] and other structural results were obtained in[1].2 Problem formulationLet n denote the vector (n1, . . . , nk). For each class k = 1, . . . , K, the inter-arrivaltime is exponentially distributed with mean value 1/λk(nk). The sojourn times ofthe accepted objects are independent and identically distributed (i.i.d.) with meanvalues 1/µk, k = 1, . . . , K. At the time of its arrival, each object is either acceptedor rejected, according to a coordinate-convex policy, defined as follows [6, p. 116].1 Note that this is different from the “generalized stochastic knapsack” considered in [6,Chapter 3].106Definition 2.1 A nonempty set Ω ⊆ ΩFR ⊂ NK0 is coordinate-convex iff it has thefollowing property: for every n ∈ Ω with nk > 0 one has n − ek ∈ Ω, where ekis a K-dimensional vector whose k-th component is 1 and the other ones are 0.The coordinate-convex policy associated with a coordinate-convex set Ω admits anarriving object iff after its insertion one has n ∈ Ω.Note that by (3) or (4), the set ΩFR is itself coordinate-convex. As there is a one-to-one correspondence between coordinate-convex sets and coordinate-convex poli-cies, in the following we use the symbol Ω to denote either a coordinate-convex setor a coordinate-convex policy.The objective to be maximized in the set P(ΩFR) of coordinate-convex subsets ofΩFR is given byJ(Ω) =∑n∈Ω(n · r)PΩ(n) , (5)where r denotes the vector (r1, . . . , rK) and PΩ(n) is the steady-state probabilitythat the current content of the knapsack is n. As Ω is coordinate-convex, one canshow that PΩ(n) takes on the product-form expressionPΩ(n) =∏Ki=1 qi(ni)∑n∈Ω∏Ki=1 qi(ni), where qi(ni) :=∏ni−1j=0 λi(j)ni!µnii. (6)Due to (6), in general the objective (5) is nonlinear. What makes the problem dif-ficult is that, given any two coordinate-convex sets Ω1, Ω2 ⊆ ΩFR, in general therelationship Ω1 ⊆ Ω2 does not imply J(Ω1) ≤ J(Ω2).3 ContributionsFor generalized stochastic knapsack problems with two classes of objects (whichmodel CAC with two classes of users), in [1] we derived for the optimal coordinate-convex policies structural properties that do not depend on the revenue ratio R :=r2/r1. In [2], instead, we obtained properties that do depend on it. In the presentwork, we develop the investigation in the following directions.(i) We analyze the optimal choices for some parameters used by a criterion pro-posed in [1] to improve certain suboptimal coordinate-convex policies. Thecriterion is based on the removal or addition of rectangular subregions nearsuitably-defined corner points.(ii) We propose a greedy algorithm of approximate solution, based on the optimalchoice of the parameters in (i). The related simulations show an improvementof the objective over the numerical results derived in [1].(iii) We address some relationships between general structural properties of theoptimal coordinate-convex policies and the greedy algorithm in (ii). In partic-ular, we extend [1, Theorem III.6] by proving that the coordinate-convex setassociated with an optimal coordinate-convex policy has a nonempty inter-section with the upper boundary (∂ΩFR)+ of the feasibility region ΩFR (see107Fig. 1), independently of the number of its corner points.(iv) We exploit another general structural property of the optimal coordinate-convex policies to initialize the greedy algorithm in (ii).(a) (b) n1n1n2n2(!"FR)+(!"FR)+1Fig. 1. The upper boundary (∂ΩFR)+ of a feasibility region ΩFR with two classes of ob-jects in the case of (a) a linearly-constrained ΩFR (stochastic knapsack) and (b) a nonlin-early-constrained ΩFR (generalized stochastic knapsack).References[1] M. Cello, G. Gnecco, M. Marchese, and M. Sanguineti. CAC with nonlinearly-constrained feasibility regions. IEEE Communications Letters, 15(4):467–469,2011.[2] M. Cello, G. Gnecco, M. Marchese, andM. Sanguineti. Structural properties ofoptimal coordinate-convex policies for CAC with nonlinearly-constrained feasi-bility regions. In Proceedings of IEEE INFOCOM (Mini-Conference), pages466–470, 2011.[3] B. C. Dean, M. X. Goemans, and J. Vondrak. Approximating the stochasticknapsack problem: The benefit of adaptivity. Mathematics of Operations Re-search, 33(4):945–964, 2008.[4] T. Javidi and D Teneketzis. An approach to connection admission control insingle-hop multiservice wireless networks with QoS requirements. IEEE Trans-actions on Vehicular Technology, 52(4):1110–1124, 2003.[5] A. J. Kleywegt and J. D. Papastavrou. The dynamic and stochastic knapsackproblem with random sized items. Operations Research, 49(1):26–41, 2001.[6] K. W. Ross. Multiservice Loss Models for Broadband Telecommunication Net-works. Springer, New York, 1995.[7] K.W. Ross and D.H.K. Tsang. The stochastic knapsack problem. IEEE Trans-actions on Communications, 37(7):740–747, 1989.108

A Generalized Stochastic Knapsack Problem with Application in Call Admission Control

IMT Institutional Repository

A Generalized Stochastic Knapsack Problem withApplication in Call Admission ControlMarco Cello, Giorgio Gnecco, Mario Marchese,Marcello SanguinetiDepartment of Communications, Computer and Systems Science (DIST), University ofGenoa - Via Opera Pia, 13 - 16145 Genova (Italy){marco.cello,giorgio.gnecco,mario.marchese,marcello}@dist.unige.itKey words: Stochastic Knapsack, Feasibility Region, Call Admission Control,Coordinate-Convex Policies.1 IntroductionIn the classical knapsack problem, a knapsack of capacity C is given, together withK classes of objects. For every k = 1, . . . , K, each object of the class k has a sizebk and an associated reward rk. The objects can be placed into the knapsack as longas the sum of their sizes does not exceed the capacity C. The problem consists inplacing the objects inside the knapsack so as to maximize the total reward.Among the various extensions to a stochastic framework available in the literature(see, e.g., [7,5,3]), we consider the stochastic knapsack problem proposed in [7].In such a model, objects belonging to each class become available randomly, ac-cording to exponentially-distributed inter-arrivals times with means depending onthe class and on the state of the knapsack. Each object has a sojourn time indepen-dent from the sojourn times of the other objects and described by a class-dependentdistribution. If put into the knapsack, an object from class k generates revenue ata positive rate rk. Let nk ≥ 0 denote the number of objects of class k that arecurrently inside the knapsack. Then one has the linear constraint∑k∈Knkbk ≤ C . (1)The problem consists in finding a policy that maximizes the average revenue, byaccepting or rejecting the arriving objects in dependence of the current state of theknapsack.The stochastic knapsack problem that we have just described has application, e.g.,in Call Admission Control (CAC) for telecommunication networks. In such a con-text, the objects are requests of connections coming from K different classes ofCTW2011, Villa Mondragone, Frascati, Italy. June 14–16, 2011users, each with a bandwidth requirement bk, k = 1, . . . , K, and a distribution forits duration. In CAC problems, often the constraint (1) arises as a linearization ofthe nonlinear constraint∑k∈Kβk(nk) ≤ C , (2)where the βk(·) are nonlinear nonnegative functions. The model in which the linearconstraint (1) is replaced by the nonlinear one (2) is known in the literature as thegeneralized stochastic knapsack problem 1 [4]. In call admission control, the coef-ficients bk of the linearized constraint are called effective bandwidths [6, Chapter1].The setsΩFR := {(n1, . . . , nK) ∈ NK0 :∑k∈Knkbk ≤ C}, (3)in the linear case, andΩFR := {(n1, . . . , nK) ∈ NK0 :∑k∈Kβk(nk) ≤ C} (4)in the nonlinear case, are called feasibility regions. In the context of admissioncontrol they model subsets of the call space {(n1, . . . , nK) ∈ NK0 }, where givenQuality of Service (QoS) constraints are satisfied.In general, finding optimal policies is a difficult combinatorial optimization taskboth for the stochastic knapsack problem [6, Chapter 4] and for the generalizedstochastic one [1,2]. The a-priori knowledge of structural properties of the (un-known) optimal policies is useful to find the solutions or, at least, good subop-timal policies. For two classes of objects and the linear constraint (1), structuralproperties were derived in [7] for the optimal policies belonging to the familyof coordinate-convex policies. Such properties restrict the K-tuple (n1, . . . , nK)to suitable subsets of the feasibility region ΩFR. Some extensions to nonlinearly-constrained feasibility regions of the structural results obtained in [7] for linearly-constrained ones were derived in [2] and other structural results were obtained in[1].2 Problem formulationLet n denote the vector (n1, . . . , nk). For each class k = 1, . . . , K, the inter-arrivaltime is exponentially distributed with mean value 1/λk(nk). The sojourn times ofthe accepted objects are independent and identically distributed (i.i.d.) with meanvalues 1/µk, k = 1, . . . , K. At the time of its arrival, each object is either acceptedor rejected, according to a coordinate-convex policy, defined as follows [6, p. 116].1 Note that this is different from the “generalized stochastic knapsack” considered in [6,Chapter 3].106Definition 2.1 A nonempty set Ω ⊆ ΩFR ⊂ NK0 is coordinate-convex iff it has thefollowing property: for every n ∈ Ω with nk > 0 one has n − ek ∈ Ω, where ekis a K-dimensional vector whose k-th component is 1 and the other ones are 0.The coordinate-convex policy associated with a coordinate-convex set Ω admits anarriving object iff after its insertion one has n ∈ Ω.Note that by (3) or (4), the set ΩFR is itself coordinate-convex. As there is a one-to-one correspondence between coordinate-convex sets and coordinate-convex poli-cies, in the following we use the symbol Ω to denote either a coordinate-convex setor a coordinate-convex policy.The objective to be maximized in the set P(ΩFR) of coordinate-convex subsets ofΩFR is given byJ(Ω) =∑n∈Ω(n · r)PΩ(n) , (5)where r denotes the vector (r1, . . . , rK) and PΩ(n) is the steady-state probabilitythat the current content of the knapsack is n. As Ω is coordinate-convex, one canshow that PΩ(n) takes on the product-form expressionPΩ(n) =∏Ki=1 qi(ni)∑n∈Ω∏Ki=1 qi(ni), where qi(ni) :=∏ni−1j=0 λi(j)ni!µnii. (6)Due to (6), in general the objective (5) is nonlinear. What makes the problem dif-ficult is that, given any two coordinate-convex sets Ω1,Ω2 ⊆ ΩFR, in general therelationship Ω1 ⊆ Ω2 does not imply J(Ω1) ≤ J(Ω2).3 ContributionsFor generalized stochastic knapsack problems with two classes of objects (whichmodel CAC with two classes of users), in [1] we derived for the optimal coordinate-convex policies structural properties that do not depend on the revenue ratio R :=r2/r1. In [2], instead, we obtained properties that do depend on it. In the presentwork, we develop the investigation in the following directions.(i) We analyze the optimal choices for some parameters used by a criterion pro-posed in [1] to improve certain suboptimal coordinate-convex policies. Thecriterion is based on the removal or addition of rectangular subregions nearsuitably-defined corner points.(ii) We propose a greedy algorithm of approximate solution, based on the optimalchoice of the parameters in (i). The related simulations show an improvementof the objective over the numerical results derived in [1].(iii) We address some relationships between general structural properties of theoptimal coordinate-convex policies and the greedy algorithm in (ii). In partic-ular, we extend [1, Theorem III.6] by proving that the coordinate-convex setassociated with an optimal coordinate-convex policy has a nonempty inter-section with the upper boundary (∂ΩFR)+ of the feasibility region ΩFR (see107Fig. 1), independently of the number of its corner points.(iv) We exploit another general structural property of the optimal coordinate-convex policies to initialize the greedy algorithm in (ii).(a) (b) n1n1n2n2(!"FR)+(!"FR)+1Fig. 1. The upper boundary (∂ΩFR)+ of a feasibility region ΩFR with two classes of ob-jects in the case of (a) a linearly-constrained ΩFR (stochastic knapsack) and (b) a nonlin-early-constrained ΩFR (generalized stochastic knapsack).References[1] M. Cello, G. Gnecco, M. Marchese, and M. Sanguineti. CAC with nonlinearly-constrained feasibility regions. IEEE Communications Letters, 15(4):467–469,2011.[2] M. Cello, G. Gnecco, M. Marchese, andM. Sanguineti. Structural properties ofoptimal coordinate-convex policies for CAC with nonlinearly-constrained feasi-bility regions. In Proceedings of IEEE INFOCOM (Mini-Conference), pages466–470, 2011.[3] B. C. Dean, M. X. Goemans, and J. Vondrak. Approximating the stochasticknapsack problem: The benefit of adaptivity. Mathematics of Operations Re-search, 33(4):945–964, 2008.[4] T. Javidi and D Teneketzis. An approach to connection admission control insingle-hop multiservice wireless networks with QoS requirements. IEEE Trans-actions on Vehicular Technology, 52(4):1110–1124, 2003.[5] A. J. Kleywegt and J. D. Papastavrou. The dynamic and stochastic knapsackproblem with random sized items. Operations Research, 49(1):26–41, 2001.[6] K. W. Ross. Multiservice Loss Models for Broadband Telecommunication Net-works. Springer, New York, 1995.[7] K.W. Ross and D.H.K. Tsang. The stochastic knapsack problem. IEEE Trans-actions on Communications, 37(7):740–747, 1989.108

An approach to connection admission control in single-hop multiservice wireless networks with QoS requirements.

Approximating the stochastic knapsack problem: The beneﬁt of adaptivity.

CAC with nonlinearlyconstrained feasibility regions.

Multiservice Loss Models for Broadband Telecommunication Networks.

Structural properties of optimal coordinate-convex policies for CAC with nonlinearly-constrained feasibility regions.

The dynamic and stochastic knapsack problem with random sized items.

The stochastic knapsack problem.

https://iris.unige.it/bitstream/11567/332855/1/CTW11.pdf

A Generalized Stochastic Knapsack Problem with Application in Call Admission Control

Abstract

Similar works

Full text

Available Versions

Archivio istituzionale della ricerca - Università di Genova

IMT Institutional Repository