Numerical illustrations 1. Experimental setting

i∈M

Θi

+ τ =

Φ

(⃗

^K

,

^B

,

G

),

⁽⁶⁰⁾

where

τ ≥

0 is the reserved bandwidth, andΦ

(⃗

^K

,

^B

,

G

)

is the total available bandwidth purchased with limited budget B for preset numbers of virtual paths_K

⃗ = (

^K1

, . . . ,

^Km

)

on the core network G. Then, the budget ratio for each traffic class i

∈

M is given below.

Definition 7. A budget ratio allocated to class i is defined as the fraction of total available bandwidthΦ

(⃗

^K

,

^B

,

G

) < ∞

allocated for class i

∈

M. That is, the budget ratio allocated for class i

∈

_{M is}

B_i, Θi

Φ

(⃗

^K

,

^B

,

G

) ,

⁽⁶¹⁾

where the maximum throughputΘi

=

K_ix_i.

If there exists sufficient bandwidth, the reserved bandwidth

τ >

0 may be shared among all the ongoing connections with budget ratio B_ifor each class i

∈

M at the online process. That is, any remaining bandwidth is shared according to the budget ratio B_i. In case there is no bandwidth reserved (

τ =

0), the allocated bandwidth x_iwill decrease if K_iincreases when those preset numbers of virtual paths, K_i′

,

^i′

̸=

i, of other classes are fixed.

Remark 9. If the maximum throughputΘiincreases, from(60), the value

∑

i′̸=iΘi′will decrease. Since those numbers K_i′

are fixed for all i^′

̸=

i, the bandwidth x_i′will decrease. That is, for certain class i^′

̸=

i, the bandwidth x_i′may be snatched by class i as the maximum throughputΘiof class i becomes larger.

7. Numerical illustrations 7.1. Experimental setting

In this section, we present numerical results to show the optimal bandwidth allocation for different traffic classes under the budget constraint. Here, we select four traffic classes as test examples from statistical data monitored at the Cooperative Association for Internet Data Analysis (CAIDA) [32]. Connections of class 4 have the highest priority, and traffic class 1 is given the lowest priority. The number of connections in the high-priority traffic class is often less than that in the low-priority class, but the traffic demand and bandwidth requirement of high-priority traffic class are always larger than those of low-priority traffic class. Those parameters are summarized inTable 1, including class weight

w

i, minimum bandwidth requirement b^min_i (Mbps), aspiration level a_i(Mbps), reservation level r_i(Mbps), the average cost c_i(cents) of one unit bandwidth through class i’s virtual paths, number of virtual paths K_i, mean occurrence rate

λ

i, the connection volume

σ

i(Mb), cost charged for using per unit of bandwidth c_i^b(cents), cost per unit of sojourn time c_i^t(cents), payoff p_i(cents) and opportunity cost q_i(cents).

7.2. Comparison between management scheme I and management scheme II

The allocated bandwidth is determined by solving Revenue Management Scheme I and Revenue Management Scheme II, respectively. Table 2shows those optimal bandwidth allocation and optimal values of Management Scheme I and

432 C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439

Table 1

Characteristics of each traffic class.

i wi b^min_i ai ri yi ci Ki λi σi c_i^t c_i^b pi qi

Budget versus optimal bandwidth allocation xi.

Budget B Scheme I Scheme II

Optimal bandwidth allocation(^x1,^x2,^x3,^x4) ^{Revenue F} Optimal bandwidth allocation(^x1,^x2,^x3,^x4) ^{Profit G}

500 (0.80, 1.00, 2.08, 2.50) 10,141 (0.80, 1.00, 2.00, 2.59) 1,412

550 (0.80, 1.00, 2.74, 2.50) 12,898 (0.80, 1.08, 2.08, 3.11) 4,480

600 (0.80, 1.00, 3.40, 2.50) 16,254 (0.80, 1.20, 2.30, 3.44) 7,187

650 (0.80, 1.00, 4.06, 2.50) 19,997 (0.80, 1.31, 2.52, 3.77) 9,646

700 (0.80, 1.00, 4.72, 2.50) 23,787 (0.80, 1.42, 2.74, 4.10) 11,900

750 (0.80, 1.00, 5.38, 2.50) 27,284 (0.80, 1.54, 2.97, 4.43) 13,980

800 (0.80, 1.00, 6.04, 2.50) 30,379 (0.80, 1.65, 3.19, 4.76) 15,911

850 (0.80, 1.00, 6.71, 2.50) 33,203 (0.82, 1.76, 3.40, 5.07) 17,714

900 (0.80, 1.00, 7.37, 2.50) 35,911 (0.87, 1.87, 3.60, 5.37) 19,412

950 (0.80, 1.00, 8.03, 2.50) 38,584 (0.92, 1.97, 3.80, 5.67) 21,018

1000 (0.80, 1.00, 8.69, 2.50) 41,247 (0.97, 2.07, 4.00, 5.97) 22,542

Table 3

Budget versus blocking probability of allocated bandwidth.

Budget B Blocking probability in Scheme I Blocking probability in Scheme II

(^P(^x1,^K1,^y1),^P(^x2,^K2,^y2),^P(^x3,^K3,^y3),^P(^x4,^K4,^y4)) (^P(^x1,^K1,^y1),^P(^x2,^K2,^y2),^P(^x3,^K3,^y3),^P(^x4,^K4,^y4))

500 (0.0020, 0.2504, 0.5389, 0.7938) (0.0020, 0.2504, 0.5553, 0.7869)

550 (0.0020, 0.2504, 0.4004, 0.7938) (0.0020, 0.1977, 0.5372, 0.7440)

600 (0.0020, 0.2504, 0.2720, 0.7938) (0.0020, 0.1311, 0.4903, 0.7174)

650 (0.0020, 0.2504, 0.1616, 0.7938) (0.0020, 0.0768, 0.4442, 0.6910)

700 (0.0020, 0.2504, 0.0795, 0.7938) (0.0020, 0.0383, 0.3989, 0.6648)

750 (0.0020, 0.2504, 0.0311, 0.7938) (0.0020, 0.0158, 0.3547, 0.6386)

800 (0.0020, 0.2504, 0.0096, 0.7938) (0.0020, 0.0054, 0.3119, 0.6127)

850 (0.0020, 0.2504, 0.0025, 0.7938) (0.0010, 0.0016, 0.2725, 0.5881)

900 (0.0020, 0.2504, 0.0006, 0.7938) (0.0002, 0.0005, 0.2368, 0.5649)

950 (0.0020, 0.2504, 0.0001, 0.7938) (0.0000, 0.0001, 0.2030, 0.5419)

1000 (0.0020, 0.2504, 0.0000, 0.7938) (0.0000, 0.0000, 0.1714, 0.5191)

Management II, where the total budget B varies from $500 to $1000. For each available budget B, according to(38), the initial solutions x⁰_i

=

BDi

/(

^ci

∑

4

i=1K_iDi

)

^{for all i}

=

1

, . . . ,

4, are given in the optimization process. In can be observed fromTable 2that those bandwidth allocations and optimal values are increasing when enlarging the available budget. From Figs. 1–3, we graphically illustrate the effect of changing budget B on optimal bandwidth allocation and optimal values of those two schemes. InFig. 1, it shows that almost all the available resources are allocated in the direction of class 3 in Management Scheme I, and the others are allocated to satisfy the minimum bandwidth requirements only. However, in Management Scheme II, it can be seen inFig. 2that the available resource is allocated to all classes proportionally. Both optimal values of Management Scheme I and Management Scheme II are increasing in the total budget B. InFig. 3, we find that there exists an inflection point such that the optimal revenues of Management Scheme I are concave up when the budget B is smaller than the inflection point, and the optimal revenue is concave down if the budget exceeds the inflection point. However, the optimal profit of Management Scheme II is expressed in logarithmic form. The marginal (optimal) profit obtained by solving Management Scheme II is decreasing with respect to the available budget B.

Next, we compare bandwidth sharing policies between Scheme I and Scheme II by showing the blocking probability and budget ratio for four traffic classes.Table 3presents those blocking probabilities determined by those optimal bandwidth allocation of two management schemes. For Management Scheme I, it can be observed fromFig. 4 that the blocking probability of class 3 is decreasing when the budget B increases from $500 to $1000 while other classes’ blocking probabilities remain unimproved. However, in Management Scheme II,Fig. 5shows that those blocking probabilities of four traffic classes are decreasing proportionally when increasing the total budget.

Given a budget B, the budget ratio for each traffic class is determined from the definition in (61)as follows: B_i

=

K_ic_ix_i

/ ∑

⁴i=1K_ic_ix_i, for i

=

1

, . . . ,

4, where bandwidth x_iis determined by solving Management Scheme I and Management Scheme II, respectively. Table 4summarizes those numerical results of the budget ratio for four traffic classes when increasing the available budget B from $500 to $1000. It can be seen from Fig. 6that most of the available budget is

C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439 433

Fig. 1. Budget versus optimal bandwidth allocation of Revenue Management Scheme I.

Fig. 2. Budget versus optimal bandwidth allocation of Revenue Management Scheme II.

Fig. 3. Budget versus optimal values of two revenue management schemes.

434 C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439

Table 4

Budget versus budget ratio for four traffic classes.

Budget B Budget ratio in Scheme I Budget ratio in Scheme II

(^B1,^B2,^B3,^B4) (^B1,^B2,^B3,^B4)

500 (15.36%, 19.50%, 31.39%, 33.75%) (15.36%, 19.50%, 30.24%, 34.90%) 550 (13.96%, 17.73%, 37.63%, 30.68%) (13.96%, 19.17%, 28.65%, 38.22%) 600 (12.80%, 16.25%, 42.82%, 28.13%) (12.80%, 19.43%, 29.03%, 38.73%) 650 (11.82%, 15.00%, 47.22%, 25.96%) (11.82%, 19.65%, 29.36%, 39.17%) 700 (10.97%, 13.93%, 50.99%, 24.11%) (10.97%, 19.84%, 29.64%, 39.55%) 750 (10.24%, 13.00%, 54.26%, 22.50%) (10.24%, 20.00%, 29.89%, 39.87%) 800 (9.60%, 12.19%, 57.12%, 21.09%) (9.60%, 20.14%, 30.10%, 40.16%) 850 (9.04%, 11.47%, 59.64%, 19.85%) (9.29%, 20.21%, 30.20%, 40.29%) 900 (8.53%, 10.83%, 61.88%, 18.75%) (9.29%, 20.21%, 30.20%, 40.29%) 950 (8.08%, 10.26%, 63.89%, 17.76%) (9.29%, 20.21%, 30.20%, 40.29%) 1000 (7.68%, 9.75%, 65.69%, 16.88%) (9.29%, 20.21%, 30.20%, 40.29%)

Fig. 4. Budget versus blocking probability determined by optimal solutions of Revenue Management Scheme I.

Fig. 5. Budget versus blocking probability determined by optimal solutions of Revenue Management Scheme II.

allocated to class 3, and the other classes only get the minimum bandwidth to satisfy the feasibility. This is because the marginal improvement of the objective function in Management Scheme I is the largest in the direction of class 3, i.e.,

∂w

3F₃

(

^x3

,

^K3

,

^y3

)/∂

^x3

≥ ∂w

iF_i

(

^xi

,

^Ki

,

^yi

)/∂

^xifor i

=

₁

,

²

,

^{4.Fig. 7}shows that all the budget is allocated to four traffic classes proportionally. The budget ratios for four traffic classes are almost invariable when varying budget B from $500 to $1000.

C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439 435

Fig. 6. Budget versus budget ratio determined by optimal solutions of Revenue Management Scheme I.

Fig. 7. Budget versus budget ratio determined by optimal solutions of Revenue Management Scheme II.

7.3. Sensitivity analysis

In this subsection, we present sensitivity analysis of the blocking probability and system utilization to illustrate numerically those monotone and convex properties of objective functions in Management Schemes I and II. First, we observe the effect of changing bandwidth x_i on the blocking probabilityP

(

^xi

,

^Ki

,

^yi

)

^{defined in}(11). To conduct the sensitivity analysis, we check the bandwidth x_ifrom 0.1 Mbps to 8 Mbps, and other parameters remain fixed as listed inTable 1. It can be seen fromFig. 8that the blocking probability is decreasing and convex when increasing bandwidth, which are consistent with those theoretical results given inPropositions 1and2.

Next, we show the effect of changing bandwidth x_ion the expected path occupancyL

(

^xi

,

^Ki

,

^yi

)

, average throughputΘi

and utilization level U_i, respectively. These three performance measures have been represented as functions of the blocking probability according to(12)–(14). FromFig. 9, we find that the expected path occupancyL

(

^xi

,

^Ki

,

^yi

)

ⁱⁿ⁽¹²⁾is a decreasing function of bandwidth x_i. In addition, it can be seen from class 1 or class 2 inFig. 9that there exists an inflection point



^xi

such that for all x_i

≤ (≥) 

^xi, the expected path occupancyL

(

^xi

,

^Ki

,

^yi

)

is concave (convex) in bandwidth x_i. Those monotone and convex properties have been summarized inProposition 3andRemark 3. Furthermore, it can be observed fromFig. 10 that average throughputΘidefined in(13)is increasing in bandwidth x_ifor four traffic classes. InFig. 11, it shows the effect of changing bandwidth on the utilization levels of preset virtual paths for four traffic classes.Proposition 3infers that those utilization levels U_idefined in(14)are decreasing when enlarging bandwidth x_i, which can be seen numerically inFig. 11.

Moreover, it can be observed clearly from class 1 or class 2 that there exists an inflection point



^xisuch that U_iis concave (convex) in bandwidth x_ifor all x_i

< (>) 

^xi.

436 C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439

Fig. 8. Blocking probabilityP(xi,Ki,yi)versus bandwidth xifor four traffic classes.

Fig. 9. Expected path occupancyL(^xi,^Ki,^yi)versus bandwidth xifor four traffic classes.

In the following, we present numerical analysis of revenue function(15)in Management Scheme I and profit function (17)in Management Scheme II, individually. Those numerical results are shown inFigs. 12and13. The results can graphically illustrate monotone and convex relationships which have been proven inTheorems 1–6.

It has been proven inTheorem 1that the objective function F_i

(

^xi

,

^Ki

,

^yi

)

of Revenue Management Scheme I is increasing in bandwidth x_iif it satisfies the inequality(18). It can be seen fromFig. 12that F₁

(

^x1

,

^K1

,

^y1

)

is increasing in bandwidth x₁for all bandwidth x₁

≥

max

{



c₁^t

σ

1

/

^cb1

,

^bmin1

} =

0

.

8 Mbps. Similarly, F₂

(

^x2

,

^K2

,

^y2

)

is increasing in bandwidth x₂for all bandwidth x₂

≥

1 Mbps, and so on. We find that the convexity of average revenue(15)fluctuates when bandwidth x_iis small corresponding to other system parameters.Proposition 2infers that, for each traffic class i, there exists a region Siof bandwidth such that the blocking probabilityP

(

^xi

,

^Ki

,

^yi

)

is convex (concave) for all x_i

∈ (̸∈)

Si, where the region Sican be constructed from the proof ofProposition 2. From numerical experiments, we find that if the budget B or bandwidth x_iis large enough, those revenue function F_i

(

^xi

,

^Ki

,

^yi

)

will become increasing and concave.

Finally, we illustrate the effect on the profit G_i

(

^xi

,

^Ki

,

^yi

)

in Revenue Management Scheme II when increasing bandwidth x_i. It can be observed fromFig. 13that the economic profit G_i

(

^xi

,

^Ki

,

^yi

)

^{defined in(17)}increases for all bandwidth x_i, which has already been proved inTheorem 3.Theorem 6infers that the profit G_i

(

^xi

,

^Ki

,

^yi

)

is concave for all x_i

≤

8 Mbps, which can be seen obviously inFig. 13.

7.4. Summary

Two revenue management schemes have been investigated theoretically and numerically to determine the amount of bandwidth required by a connection for each traffic class. Given network users’ willingness-to-pay and other system parameters, our aim is to determine the bandwidth allocation that maximizes the average revenue/profit for the ISP under the budget constraint.

C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439 437

Fig. 10. Average throughputΘiversus bandwidth xifor four traffic classes.

Fig. 11. Utilization level Uiversus bandwidth xifor four traffic classes.

Fig. 12. Average revenue Fi(^xi,^Ki,^yi)versus bandwidth xifor four traffic classes.

438 C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439

Fig. 13. Economic profit Gi(xi,Ki,yi)versus bandwidth xifor four traffic classes.

In Management Scheme I, almost all the resources are allocated to only one class whose marginal revenue is the largest, and the remainder are allocated to other classes to meet their feasibility only. That is, most of the available budget is allocated to certain traffic class i with the largest marginal improvement

∂w

iF_i

(

^xi

,

^Ki

,

^yi

)/∂

^xi in Management Scheme I. Network managers may apply Management Scheme I to allocate limited resources among competing classes in order to maximize the weighted sum of average revenue. On the other hand, by solving Management Scheme II, all resources are allocated proportionally to four traffic classes. With the help of the utility function in(16), we can achieve the proportional fairness by allocating bandwidth through Management Scheme II.

To investigate these two bandwidth allocation policies, monotone and convex properties of the revenue/profit function as well as the blocking probability have been proven theoretically in previous sections and illustrated numerically in this section. Those phenomena in numerical experiments are consistent with theoretical results. In practice, those results may help network managers to determine their optimal/acceptable bandwidth allocation according to one of those two management schemes.

8. Conclusions

In this paper, we consider the revenue management problems on communication networks with multi-class traffic under the budget constraint. Two revenue management schemes have been investigated through the monotone and convex properties of the blocking probability and expected path occupancy of connections. We analyze the sensitivity of the blocking probability to model parameters, where the parameters change one-at-a-time. Under general assumptions, we have proved that the blocking probability is directionally (i) decreasing in bandwidth, (ii) convex in bandwidth for specific regions, (iii) increasing in traffic demand, and (iv) decreasing in the number of virtual paths. We also demonstrate the monotone and convex relations among the expected path occupancy and those model parameters. Furthermore, we prove that for a fixed number of virtual paths, the blocking probability is increasing and convex in traffic intensity for specific regions.

The optimality conditions are derived to obtain an optimal bandwidth allocation for two revenue management schemes.

A solution algorithm is also developed to allocate limited budget among competing traffic classes. We have conducted the sensitivity analysis of the average revenue function and the economic profit function for a given traffic class by changing bandwidth allocation, traffic demand and the available number of virtual paths respectively. Those results have also been verified with numerical examples interpreting the blocking probability, utilization level, average revenue, etc. The relationship between blocking probability and bandwidth allocation can help network managers to design network pricing mechanisms for sharing bandwidth in terms of blocking/congestion costs.

The contribution of the current paper is the analysis of those monotone and convex relations among model parameters and performance measures of interest. The results of this work may be helpful in the operational processes involved in the efficient set-up and usage of a core network under the budget constraint, e.g., network design and provisioning purposes.

One application of the relationship between blocking probability and bandwidth allocation may be referred to as designing network pricing mechanisms for sharing bandwidth in terms of blocking/congestion costs, whose examples were given by Yacoubi et al. [4] and Anderson et al. [19], etc. The closed-form expression of the blocking probability in terms of bandwidth can also be used to investigate the optimal buffer size in capacitated communication systems so that the blocking probability is kept below a specific threshold [15]. Another application of this work is used to consider the admission control in networks under different bandwidth sharing policies including throughput maximization, max–min fairness, proportional fairness and balanced fairness, etc. Interested readers may refer to Bonald et al. [5], Nilsson and Pióro [33], Jordan [34], etc.

C.-H. Wang, H.P. Luh / Computers and Mathematics with Applications 62 (2011) 419–439 439

In addition, we present three elasticities to investigate the effect of changing model parameters on the average revenue in analysis of economic models. The sensitivity results derived here could be used to guide development of congestion-based pricing of network resources, and to adjust bandwidth in the optimal proportion in response to changes in desired levels of blocking probability. Future work will be conducted in the direction of further investigation for the network revenue management schemes. Much additional work would have to be done in the future to make such an approach practical, e.g., design of reservation protocols, scheduling policies, measurement algorithms, and feedback algorithms to guarantee convergence.

Acknowledgements

The authors thank two anonymous referees for their valuable and constructive comments. This research was supported in part by National Science Council, Taiwan, R.O.C., under grant number NSC-98-2221-E-004-001-MY2.

Appendix. Proofs of propositions

Detailed proofs of propositions and corollaries can be found online atdoi:10.1016/j.camwa.2011.05.024.

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在文檔中 於預算控制下計算M/G/K/K 和 GI/M/K/K 的滿載機率 (頁 38-64)