Optimal Code Assignment and Call Admission Control for OVSF ...

Optimal Code Assignment and Call Admission Control for OVSF-CDMA Systems Constrained by Blocking Probabilities ∗ Department

Jun-Seong Park∗ , Lei Huang† , Daniel C. Lee∗ , and C.-C. Jay Kuo∗ of Electrical Engineering, Integrated Media Systems Center and Communication Sciences Institute University of Southern California, Los Angeles, CA 90089, USA E-mail: [email protected], [email protected], [email protected] † Department of Electrical Engineering and Computer Science Loyola Marymount University, Los Angeles, CA 90045, USA E-mail: [email protected]

Abstract— The tradeoff between the throughput and the blocking probability of various code assignment schemes for OVSF-CDMA systems is investigated, and the optimal fixed and dynamic code assignment schemes that maximize the average data throughput under a blocking probability constraint are proposed in this research. The optimal schemes are obtained using constrained optimization through the Markov Decision Process and linear programming. The performance of each scheme is numerically evaluated. Numerical results demonstrate that the proposed new schemes can achieve satisfactory balance between the average throughput and the blocking probability.

I. I NTRODUCTION The orthogonal-variable-spreading-factor (OVSF) codes are employed in wideband CDMA systems to support different data rates of multimedia services [1] [2]. In OVSF-CDMA systems, a set of orthogonal codes with different lengths are generated using different spreading factors. A higher data rate can be achieved using a lower spreading factor. Code assignment and reassignment schemes for OVSF-CDMA systems have been extensively investigated for last several years [3]. The algorithmic complexity and the maximal throughput have been studied for both fixed and dynamic code assignment schemes [4]. The basic network model assumed here is tied to a circuitswitched architecture in which each user consumes one code as a communication channel and transmits data at a requested fixed rate during call duration. Even though the packetswitched network architecture nowadays expands quickly in wireless networks, the circuit-switched data transmission is still widely being used in the deployed cellular systems [1]. Fixed code assignment (FCA) schemes are designed to avoid code reassignment as proposed in [5], thus reducing the implementational complexity. However, under FCA, an incoming call may be rejected because its data rate cannot be supported by any available code of its class, even if the total available system capacity can accommodate it. Dynamic code assignment (DCA) schemes improve throughput efficiency at the cost of a much higher complexity [5]. The optimal FCA with fixed set partitioning and the optimal DCA with call admission control schemes were proposed in [4] to maximize the average throughput. Generally speaking, there are two performance metrics for a code assignment scheme, i.e. the throughput and the blocking probability. Most previous work focused either one of the two. It is clear that the solution that gives the highest throughput is in general not the same as the one that gives the lowest blocking probability. For example, consider a system supporting two

service classes: class I with a low transmission rate and a high arrival rate, and class II with a high transmission rate and a low arrival rate. A code assignment scheme that serves class I only will optimize the blocking probability, while one that serves class II only will optimize the average throughput. Hence, it is interesting to study the tradeoff between these two metrics in association with different code assignment schemes. In this research, we attempt to integrate the two performance metrics into one framework of constrained optimization. That is, we seek to maximize the average throughput under the blocking probability constraint for both FCA and DCA schemes. The blocking probability constraint is imposed implicitly through a concept called the constraint strength, which will be detailed in Sec. II-A. We propose new constrained FCA and DCA schemes to reach a good balance between the two performance metrics. It is demonstrated by experimental results that we are able to achieve consistent performance improvement over a wide range of traffic load. The rest of this paper is organized as follows. We propose the optimal fixed and dynamic code assignment schemes in Secs. II and III, respectively. Numerical results are given in Sec. IV. Concluding remarks and future research work are given in Sec. V. II. O PTIMAL FCA WITH B LOCKING P ROBABILITY C ONSTRAINT The FCA scheme partitions the whole set of OVSF codes into mutually exclusive groups of codes and uniquely assigns a group to each service class determined by the service data rate [4]. Thus, the number of available codes for each service class is fixed, and no code reassignment is allowed. A. Problem Formulation We consider M service classes of calls arriving according to independent Poisson processes. Calls of class-k requesting the transmission rate of 2k R arrive at rate λk , k = 0, 1, 2, . . . , M − 1, where R is the basic data transmission rate (or the codes with the highest spreading factor in the OVSF code tree), and k indicates the service class corresponding to the transmission rate of 2k R. The call duration for all classes is assumed to be exponentially distributed with mean 1/µ. In the circuitswitched network model, the Poisson arrivals and the exponential call duration are reasonable assumptions and analytically tractable. The traffic load requesting the transmission rate of k is defined as ρk = λk /µ. We also define Cmax as the total number of the highest-spreading-factor codes in a given

OVSF code tree. Assuming an ideal code-limited (single-cell) scenario, the system can support a maximum information bandwidth of RCmax . Furthermore, we use Gk to denote the total number of codes in a group that supports the data rate of class k. Then, there exists a finite number of code partitioning, represented by G
(G0 , G1 , . . . , GM −1 ), that satisfies the maximum capacity constraint G0 + 2G1 + 4G2 + · · · + 2M −1 GM −1 = Cmax . Each group of codes is uniquely assigned to one of the service classes. Based on the statistics of incoming traffic, we want to find the optimal partition G∗ = (G∗0 , G∗1 , . . . , G∗M −1 ) that maximizes the average throughput under a blocking probability constraint. The blocking probability Pk of the code group corresponding to class-k can be computed using Erlang’s formula [6]: ρGk /Gk ! Pk =
Gkk . n n=1 ρk /n!

(1)

Then, the average blocking probability PF (G) of the entire system with a fixed set partition G = (G0 , G1 , . . . , GM −1 ) becomes M −1  λk Pk , (2) PF (G) = λ k=0
M −1 where λ = i=0 λi . We can also obtain the average throughput TF (G) of the system as TF (G) =

M −1  k=0

(1 − Pk )

λk k (2 R). µ

(3)

We extend the formulation in [4] by adding a blocking probability constraint to the optimization problem. That is, we seek the optimal partition of the OVSF code tree, G∗TB = (G∗0 , G∗1 , . . . , G∗M −1 ), that maximizes the average throughput TF under a blocking probability constraint. In terms of mathematics, we have a constrained optimization problem of the following form: G∗TB = argmax TF (G) G subject to PF (G) ≤ δB · PF (G∗B ),

(4) G∗B

where δB ≥ 1 is called the constraint strength, and is the optimal partition that minimizes the blocking probability without any constraint, which gives a target for the desired blocking probability. This constrained optimization problem can be solved by exhaustive search, which may not be practical in real world applications. The complexity reduction technique will be discussed in a separate report. B. Discussion on Blocking Probability Constraint Before we move to DCA, let us discuss the meaning of the blocking probability constraint in depth. Please note that the target provides the lower bound on the blocking probability that any FCA can achieve. In other words, we cannot lower it furthermore with or without constraints. By imposing the constraint in (4), we allow the blocking probability to fall in the range of [PF (G∗B ), δB · PF (G∗B )] . (5) By choosing a larger value of δB , we are willing to sacrifice the blocking probability in exchange for a larger throughput,

and vice versa. For a sufficiently large value of δB , the constraint may not have any actual effect. Then, the constrained optimization problem is reduced to an unconstrained problem. We use G∗T to denote the optimal partition that maximizes the average throughput without any constraint. Since G∗B is a function of the traffic load, the permissible blocking probability range as given in (5) is also a function of the load of the system. It is worthwhile to point out that there exist many different ways to impose the constraint on the blocking probability. For example, we can impose an upper bound on the blocking probability directly, say, PF (G) ≤ P0 ,

(6)

where P0 is a constant less than 1. The simple constraint (6) is attractive for each individual user, since it provides an explicit quality of service (QoS) measure for the user. However, this constraint may not be as simple as it appears. By following this formulation, it is often that we are not able to find any solution. This can be explained as follows. The least achievable blocking probability varies significantly with the traffic condition and the number of service classes. It is very difficult to specify a fixed blocking probability level for all service classes under all traffic conditions a priori. From the user viewpoint, we may call (5) and (6) implicit and explicit blocking probability constraints, respectively. In this research, we focus on the implicit constraint that helps service providers as well as users to achieve better throughputblocking probability tradeoff over a wide range of the traffic load. We will deal with the optimization problem with an explicit constraint more thoroughly in the future. III. O PTIMAL DCA WITH B LOCKING P ROBABILITY C ONSTRAINT In this section, we examine the dynamic code assignment (DCA) scheme, which allows code reassignment. The DCA scheme proposed by Minn and Shiu [5] will be first reviewed in Sec. III-A. Then, we formulate and derive a DCA scheme with the blocking probability constraint from Secs. III-B to III-E. A. DCA with Greedy CAC The DCA scheme proposed in [5] is characterized by code reassignment using a greedy call admission control (CAC) policy. It rearranges the OVSF code assignment to calls in progress to admit a newly arriving call whenever there is enough capacity available in the code tree. Based on the same traffic model described in Sec. II-A, the average throughput of DCA can be found by [4]  To (π) = Ωo (i)π(i) with probability 1, (7) i∈Io (Cmax ) where Io (Cmax ) is the state space of the Markov chain corresponding to the dynamics of DCA, Ωo (i) is the total code rate served in state i, and π = {π(i)} is the steady state probability of the Markov chain, i.e. the fraction of time spent

in state i in the long run. From the same Markovian analysis, we can find the blocking probability of DCA as ⎡ ⎤ Po (π) =

M −1  j=0

λj λ

⎢ ⎢ ·⎢ ⎣



⎥ ⎥ π(i)⎥ w.p.1. ⎦

(8)

i∈Io (Cmax ) Ωo (i)>(Cmax −2j ))

B. Problem Formulation We formulate the optimal DCA problem under the blocking constraint in the framework of the Continuous Time Markov Decision Process (CTMDP) [7]. In addition to the arrival and departure model, an action indicating accept or reject is introduced upon each arrival of call requests. Our design objective is to maximize the average throughput under a blocking probability constraint. We define the state of the CTMDP in such a way that the state at each epoch can indicate whether that epoch is initiated by a call arrival or a call departure. The state space, denoted by ID , is the union of two sets, i.e. the set of states indicating that the epoch begins with an arrival and the set of states indicating that the epoch begins with a departure. A state in the latter set is denoted by vector (k), where each component is an integer referring to the number of calls in progress in the corresponding class. A state in the former set is denoted by (k, j) = ((k0 , k1 , . . . , kM −1 ), j), which means that the epoch has been initiated by an arrival of class-j call, and that there have been ki calls in progress of class i immediately prior to this arrival. Note that the current number of class-j calls in progress for k in state (k, j) depends on the admission control action taken at the beginning of the epoch. Then, our extended state space ID (Cmax ) has the form

M −1 



i 2 ki ≤ Cmax , 0 ≤ j ≤ (M − 1) ID (Cmax ) = (k, j)

i=0

M −1 



∪ (k)

(9) 2i ki ≤ Cmax ,

i=0

where k
(k0 , k1 , . . . , kM −1 ). For each state (k, j) in the first set, the corresponding decision set is D(k, j) = {a, r}, where a denotes accept and r denotes reject. Note that the component of k corresponding to class j in state (k, j) increases by 1 if an action of accept is taken at the beginning of the epoch. Otherwise, the current number of class-j calls in progress in state (k, j) remains the same as the one observed at the beginning of the epoch. For each (k) in the second set, there is no action to be taken and D(k) = φ is set. C. Throughput and Blocking Probabilities The total throughput of the system at each time can be described by the combination of the CTMDP state and the action (admission or rejection) taken at the beginning of the epoch. We introduce a function, Ωe (x(t), u(t)), of the stateaction pair, which represents the total data rate at time t, where x(t) is the state of the CTMDP and u(t) is the action taken at the beginning of time t’s epoch. To be more specific, u(t) is defined as follows. For time t such that x(t) = (k, j) ∈ ID (Cmax ), u(t) = a if the decision made at the beginning

of the epoch is accept, and u(t) = r if the decision made at the beginning of the epoch is reject. For time t such that x(t) = (k) ∈ ID (Cmax ) at the beginning of its epoch, no decision needs to be made and thus u(t) is null. When action u(t) is chosen at time t (0 ≤ t < ∞), the average throughput TD (u) is the total data rate averaged over the entire time horizon, i.e. 1 T TD (u) = lim Ωe (x(t), u(t)) dt, (10) T →∞ T 0 where Ωe (x(t), u(t)) is the total date rate in state x(t) when decision u(t) was made at the beginning of the state epoch. It is given by Ωe (x(t), u(t)) ⎧  M −1 ⎪ ) ×R x(t) = (k, j), u(t) = a ⎨(k+ej )·(1, 2, . . . , 2 = k · (1, 2, . . . , 2M −1 ) ×R x(t) = (k, j), u(t) = r (11) ⎪ ⎩k · (1, 2, . . . , 2M −1 ) ×R x(t) = (k), u(t) = Null, where ej
(eR , e2R , . . . , e2M −1 R ) is a unit vector indexed by service class j ∈ {R, 2R, . . . , 2M −1 R}, for which eh = 1 if h = j and eh = 0, otherwise, and · denotes the dot product. With an optimal CAC scheme, a call blocking occurs due to one of the following two reasons: 1) The system is not able to accommodate a new call request due to the capacity limit. 2) It is intentionally rejected to accommodate a later call arriving requesting a higher rate. The entire blocking probability PD that incorporates the above two cases becomes ⎡ ⎤ M −1   λj ⎣ ˜ = PD (π) · π ˜ ((k, j); r)⎦ , (12) λ j=0 (k,j)∈ID (Cmax ) ˜
{˜ where π π (i, d)|i ∈ ID (Cmax ), d ∈ D(i)} is the set of steady state probabilities under a stationary CAC, and π ˜ ((k, j); r) is the steady state probability of the states that an arriving class-j call request is rejected by the CAC policy. D. Uniformization and Transition Probabilities The probabilistic behavior of the above CTMDP can be analyzed more conveniently in the discrete-time domain through uniformization [7]. It also allows the implementation of a numerical solution method such as linear programming. We use the minimum expected time between state transitions,
M −1 η(Cmax ) = [ j=0 λj + Cmax µ]−1 , for system capacity Cmax . With uniformization, we use p [(k, j), i; d] to denote the transition probability from state (k, j) to i if decision d is made, where d ∈ {a, r}; and p [(k), i] to denote the transition probability from state (k) to i, where there is no decision to be made. Then, all possible transition probabilities of the uniformized CTMDP are described as follows. For states (k, j) ∈ ID (Cmax ) taking accept for j ∈ {0, 1, . . . , M − 1}, p [(k, j), i; a] ⎧ λζ · η i = (k + ej , ζ) ⎪ ⎪ ⎪ ⎨kξ µ · η i = (k + ej − eξ ) 

= (13) ⎪ 1− λ +µ k · η i = (k + ej ) ⎪ n n m∈B(k) m ⎪ ⎩ 0 otherwise,

where i ∈ ID (Cmax ), ζ, ξ ∈ {0, 1, . . . , M − 1}, and B(k) = {m|k + ej − em ∈ ID (Cmax )}. For states k taking no action and (k, j) ∈ ID (Cmax ) taking reject for j ∈ {0, 1, . . . , M − 1},

•

p [(k, j), i; r] = p [(k), i] ⎧ λζ · η ⎪ ⎪ ⎪ ⎨kξ µ · η  

= ⎪ ·η 1− λ +µ k n m ⎪ n m∈B(k) ⎪ ⎩ 0

•

i = (k, ζ) i = (k − eξ )

• •

•

(14)

•

where i ∈ ID (Cmax ), ζ, ξ ∈ {0, 1, . . . , M − 1}, and B(k) = {m|k − em ∈ ID (Cmax )}.

•

i = (k) otherwise,

E. Linear Programming Solution A stationary randomized CAC can be described by β = {β(i, d) : d ∈ D(i), i ∈ ID (Cmax )} with the interpretation that, if the system enters state i, action d is to be chosen with probability βi (d). Determining the CAC policy that maximizes the average throughput with a blocking probability constraint is now formulated as the following linear programming problem [8] [9],   maximize Ωe (i, d)˜ π (i, d) (15) i∈ID (Cmax ) d∈D(i)

˜ ∗B π

(16)

(17)

(18)

∗ {πB (i, d)}

where δB is the constraint strength,
is a set of the steady state probabilities under the optimal stationary CAC that minimizes the blocking probability of the entire system without any constraint, and π ˜ (i, d) is the steady state probability that the system is in state i while decision d is chosen in the randomized policy. ˜ ∗TB
{˜ The optimal π πT∗B (i, d)} results in the optimal randomized CAC policy given by β ∗TB , where π ˜T∗B (i, d) ˜T∗B (i, z) z∈D(i) π

βT∗B (i, d) =


B. Performance Comparison of FCA Schemes Let us first compare the performance of two FCA schemes (with and without the blocking probability constraint) and use the DCA scheme with greedy CAC as the reference scheme. The average throughput and the corresponding blocking probability with a constraint strength of δB = 1.05 are shown in Figs. 1 and 2. Similar to that reported in [4], we observe that DCA with greedy CAC has good performance in terms of the throughput as well as the blocking probability when the system is lightly loaded and dominated by low rate users. Furthermore, as shown in Figs. 1 and 2, the optimal FCA under a blocking probability constraint (with δB = 1.05) yields the throughput performance very close to that of FCA optimized solely for the average throughput while maintaining the blocking probability almost equivalent to that of FCA optimized solely for the blocking probability. Generally speaking, we can obtain a trade-off between the average throughput and the blocking probability by fine-tuning the constraint strength. Finally, it is interesting to point out that the constrained optimal FCA starts to outperform DCA with greedy CAC in the blocking probability when the traffic load increases. This shows the limit of the DCA scheme with greedy CAC. Total average throughput for Cmax = 32 and TP = (0.4,0.4,0.1,0.1) 30

(19)


when ˜T∗B (i, z) > 0, otherwise, βT∗B (i, d) = 0. z∈D(i) π ˜ ∗T that maximizes the average In addition, we can obtain π throughput under no constraint for comparison. IV. N UMERICAL R ESULTS A. Experiment Setup The numerical analysis uses the following parameters. • Call arrival process is Poisson with mean arrival rate λ = λ0 + λ1 + λ2 + λ3 = 1 ∼ 16(32) calls/unit time. • Call duration is exponentially distributed with a mean value of 1/µ = 1 unit of time. The performance varies with ρk = λk /µ so setting a constant µ is reasonable.

25 Total average throughput

˜ ≤ δB · PD (π ˜ ∗B ), δB ≥ 1 subject to PD (π)    π ˜ (j, d) = p[i, j; d]˜ π (i, d) i∈ID (Cmax ) d∈D(i) d∈D(j ) for j ∈ ID (Cmax ),   π ˜ (i, d) = 1, i∈ID (Cmax ) d∈D(i) π ˜ (i, d) ≥ 0 for i ∈ ID (Cmax ), d ∈ D(i).

Traffic load is ρ = λ/µ = 1 ∼ 16(32). System capacity = Total number of the leaf codes Cmax = 16, 32. Base data transmission rate normalized to R = 1.0. Total number of service classes M = 4 with service rates equal to R, 2R, 4R and 8R. Traffic profile TP
(ρ0 /ρ, ρ1 /ρ, ρ2 /ρ, ρ3 /ρ) = (0.4, 0.4, 0.1, 0.1) is used. Constraint strength δB = 1.04 or 1.05. By adjusting δB , the optimization becomes biased toward the average throughput or the blocking probability. An ideal code-limited scenario is assumed. No interference-limited scenario is considered to define the system capacity.

20

15

10 FCA optimized for throughput w/o constraint FCA optimized for throughput w/ blocking probability constraint δB = 1.05

5

DCA w/ Greedy CAC 0

0

5

10

15 20 Traffic load (ρ)

25

30

35

Fig. 1. Comparison of the average throughput when Cmax = 32, δB = 1.05 and TP= (0.4, 0.4, 0.1, 0.1).

Total blocking probability for C

max

Total blocking probability for C

= 32 and TP = (0.4,0.4,0.1,0.1)

max

= 16 and TP = (0.4,0.4,0.1,0.1)

0.5 FCA optimized for blocking probability w/o constraint FCA optimized for throughput w/ blocking probability constraint δB = 1.05 DCA w/ Greedy CAC

0.4 Total blocking probability

Total blocking probability

0.5

DCA w/ Greedy CAC DCA optimized for blocking probability DCA optimized for throughput w/ blocking probability constraint δB=1.04

0.45

0.4

0.3

0.2

0.35 0.3 0.25 0.2 0.15 0.1

0.1

0.05 0

0

5

10

15 20 Traffic load (ρ)

25

30

35

Fig. 2. Comparison of the blocking probability when Cmax = 32, δB = 1.05 and TP= (0.4, 0.4, 0.1, 0.1).

C. Performance Comparison of DCA Schemes Next, we compare the performances of three DCA schemes. They are: (1) the one optimized for the least blocking probability, (2) the one optimized for the maximum throughput, and (3) the one optimized for the maximum throughput under a blocking probability constraint with a constraint strength of δB = 1.05. We also use DCA with greedy CAC as a benchmark. The results are shown in Figs. 3 and 4. We see that the throughput performance of the proposed DCA scheme optimized under a blocking probability constraint is close to that of the one optimized solely for the average throughput as shown in Fig. 3 especially when the traffic load is high. The blocking probability of the proposed constrained DCA scheme is very close to that of DCA optimized solely for the blocking probability as shown in Fig. 4. The blocking probability of the DCA scheme with greedy CAC is much higher. In other words, we pay little price by imposing the blocking probability constraint. Again, we are able to obtain good trade-off between the throughput and the blocking probability. Total average throughput for Cmax = 16 and TP = (0.4,0.4,0.1,0.1) 14 13

Total average throughput

12 11 10 9 8 7 DCA w/ Greedy CAC DCA optimized for throughput DCA optimized for throughput w/ blocking probability constraint δB=1.04

6 5 4

2

4

6

8 10 Traffic load (ρ)

12

14

16

Fig. 3. Comparison of the average throughput when Cmax = 16, δB = 1.04 and TP = (0.4, 0.4, 0.1, 0.1).

0

2

4

6

8 10 Traffic load (ρ)

12

14

16

Fig. 4. Comparison of the blocking probability when Cmax = 16, δB = 1.04 and TP = (0.4, 0.4, 0.1, 0.1).

V. C ONCLUSION AND F UTURE W ORK The two performance metrics, i.e. throughput and the blocking probability, associated with OVSF code assignment schemes were examined under a unified framework via constrained optimization in this work. The optimal FCA and DCA schemes that maximize the average throughput under a blocking probability constraint were proposed. Several FCA schemes (optimized solely for throughput or for the blocking probability) and DCA with greedy CAC were used as benchmarks for performance comparison. It was demonstrated that the proposed constrained FCA and DCA schemes provide an average data throughput close to that of the one optimized for the throughput without any constraint while maintaining the blocking probability close to that optimized for the blocking probability without any constraint, thus achieving good balance between the revenue reward and the quality of service. The optimal FCA and DCA schemes (with or without a constraint) both demand a high computational complexity. In the near future, we will investigate a suboptimal code assignment scheme to reduce the complexity significantly while maintaining satisfactory throughput and blocking probability performance. R EFERENCES [1] F. Adachi, M. Sawahashi, and H. Suda, “Wideband DS-CDMA for nextgeneration mobile communications systems,” IEEE Communications Mag., Sep. 1998. [2] E. H. Dinan and B. Jabbari, “Spreading codes for direct sequence CDMA and wideband CDMA cellular networks,” IEEE Communications Mag., Sep. 1998. [3] Y. C. Tseng and C. M. Chao, “Code placement and replacement strategies for wideband CDMA OVSF code tree management,” IEEE Trans. on Mobile Computing, vol.1, No.4, Dec. 2002. [4] J.-S. Park and D. C. Lee, “Enhanced fixed and dynamic code assignment policies for OVSF-CDMA systems,” Int. Conf. Wireless Networks 2003, Las Vegas, June, 2003. [5] T. Minn and K.-Y. Siu, “Dynamic assignment of orthogonal variablespreading-factor codes in W-CDMA,” IEEE J. Slect. Areas Commun., vol.18, No.8, Aug. 2000. [6] L. Kleinrock, Queueing Systems, Vol. I: Theory. New York: Wiley, 1975. [7] M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming. New York: Wiley, 1994. [8] E. A. Feinberg, “Contstrained semi-Markov decision processes with average rewards,” ZOR-Mathematical Methods Operations Res., vol.39, pp. 257-288, 1994. [9] S. M. Ross, Applied probability models with optimization applications, San Francisco: Holden-Day, 1970.