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arxiv: 2601.15103 · v1 · submitted 2026-01-21 · 💻 cs.NI · econ.TH

Economic feasibility of virtual operators in 5G via network slicing

Erwin J. Sacoto-Cabrera , Luis Guijarro , Jose R. Vidal , Vicent Pla This is my paper

Pith reviewed 2026-05-16 11:57 UTC · model grok-4.3

classification 💻 cs.NI econ.TH
keywords 5G network slicingvirtual operatorseconomic incentivesgame theorydiscriminatory processor sharingbusiness modelssubscription ratesmonopolistic and strategic models
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The pith

In 5G network slicing, pricing can be set so the infrastructure owner has incentives to support a virtual operator in both monopolistic and strategic business models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes two ways a network operator and a virtual operator can share 5G infrastructure. In the monopolistic model the network operator serves users from both operators. In the strategic model the virtual operator serves its own users and pays the network operator a per-subscriber fee. A discriminatory processor sharing queue models the sliced network performance while game theory models how users decide to subscribe and how operators set prices. The analysis shows the network operator can be compensated enough to participate in either arrangement, and users subscribe at higher rates under the strategic model.

Core claim

The paper establishes that appropriate pricing strategies in both the monopolistic model, where the network operator serves users from both operators, and the strategic model, where the virtual operator pays a per-subscriber fee to use the infrastructure, provide the necessary incentives for the network operator to participate, resulting in higher user subscription rates under the strategic model.

What carries the argument

A discriminatory processor sharing queue that models performance across network slices, combined with game-theoretic modeling of user subscription utilities and operator revenues to find equilibrium prices and subscription decisions.

If this is right

  • The network operator will agree to serve the virtual operator's users if compensated through appropriate pricing in the monopolistic model.
  • In the strategic model the network operator will allow the virtual operator to provide service over its infrastructure for a per-subscriber fee.
  • Users achieve higher subscription rates when the virtual operator directly serves them and pays the fee than when the network operator serves everyone.
  • Both models demonstrate economic feasibility for virtual operators using shared 5G infrastructure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Regulators could apply the same pricing logic to encourage voluntary infrastructure sharing without mandates.
  • Real deployments would need to test whether signaling and management overheads reduce the modeled revenue gains enough to break the incentive alignment.
  • Adding competition among multiple virtual operators could change the equilibrium subscription rates and fees found in the two-operator game.

Load-bearing premise

That user subscription choices and operator pricing can be captured accurately by the discriminatory processor sharing queue and game-theoretic revenue functions without unmodeled effects such as regulatory rules or extra technical costs changing the incentives.

What would settle it

A simulation or measurement in a real 5G sliced network where the network operator's calculated revenue gain is insufficient to accept the virtual operator's users in either model, or where user subscription rates do not rise under strategic pricing.

Figures

Figures reproduced from arXiv: 2601.15103 by Erwin J. Sacoto-Cabrera, Jose R. Vidal, Luis Guijarro, Vicent Pla.

Figure 1
Figure 1. Figure 1: Network Model by [6, 11, 7] in the context of the economic analysis of the internet service under the DiffServ paradigm. Furthermore, a DPS discipline is chosen in this work in order to model the sharing of the network capacity that is enabled by network slicing. Specifically, this DPS queue manages two priorities and each subscriber base receives service for its packets at an instantaneous rate proportion… view at source ↗
Figure 2
Figure 2. Figure 2: Wardrop equilibrium cases/regions for γ = 1/10 and α1 = α2 = 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Case I Case II Case III Case IV [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Wardrop equilibrium cases/regions for γ = 1/10 α1 = 0.2 and α2 = 0.8 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Wardrop equilibrium cases/regions for γ = 1/2 and α1 = α2 = 0.8 Case I Case II Case III Case IV 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.2 1.0 0.8 0.6 0.4 0.2 0.0 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Wardrop equilibrium cases/regions for γ = 1/2, α1 = 0.2 and α2 = 0.8 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Price p ∗ 1m as a function of 1 − γ for different values of the common α (monopolistic) 4.1. Monopolistic business model The equilibrium in the monopolistic scenario, where the NO operates the network and provides service to its own subscriber base and to the VO’s subscriber base, is presented and discussed. The NO’s subscriber base is assigned a priority 1−γ in the use of the network capacity, while the V… view at source ↗
Figure 7
Figure 7. Figure 7: Number of subscribers n ∗ 1m as a function of 1 − γ for different values of the common α (monopolistic) Two different evolutions of n ∗ 1m with 1 − γ can be distinguished, which depend on α. For low sensitivity users (i.e., low values of α, such as 0.2), the number of subscribers n ∗ 1m increases as their priority (1 − γ) increases. However, for high sensitivity users (i.e., α ≥ 0.4), the number of subscri… view at source ↗
Figure 8
Figure 8. Figure 8: Total number of subscribers as a function of [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: NO’s profit Π∗m as a function of γ for different values of the common α (monopolistic) 15 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Price p ∗ 1m as a function of 1 − γ for different values of α1 and α2 (monopolistic) [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Price p ∗ 2m as a function of γ for different values of α1 and α2 (monopolistic) 17 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Number of subscribers n ∗ 1m as a function of 1 − γ for different values of α1 and α2 (monopolistic) [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Number of subscribers n ∗ 2m as a function of γ for different values of α1 and α2 (monopolistic) 18 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Total number of subscribers as a function of [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: NO’s profits Π∗m as a function of γ for different values of α1 and α2 (monopolistic) 19 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Price p ∗ 1 as a function of 1 − γ for different values of δ (strategic) [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Price p ∗ 2 as a function of γ for different values of δ (strategic) 21 [PITH_FULL_IMAGE:figures/full_fig_p021_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Number of subscribers n ∗ 1 as a function of 1 − γ for different values of δ (strategic) [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Number of subscribers n ∗ 2 as a function of γ for different values of δ (strategic) 22 [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Total number of subscribers as a function of [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: NO’s profit Π∗ 1 as a function of 1 − γ for different values of δ (strategic) [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: VO’s profit Π∗ 2 as a function of γ for different values of δ (strategic) 24 [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Price p ∗ 1 as a function of 1 − γ for different values of α2 (strategic scenario, α1 = 0.6, δ = 0.15) [PITH_FULL_IMAGE:figures/full_fig_p026_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Price p ∗ 2 as a function of γ for different values of α2 (strategic scenario, α1 = 0.6, δ = 0.15) 26 [PITH_FULL_IMAGE:figures/full_fig_p026_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Number of subscribers n ∗ 1 as a function of 1 − γ for different values of α2 (strategic scenario, α1 = 0.6, δ = 0.15) [PITH_FULL_IMAGE:figures/full_fig_p027_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Number of subscribers n ∗ 2 as a function of γ for different values of α2 (strategic scenario, α1 = 0.6, δ = 0.15) 27 [PITH_FULL_IMAGE:figures/full_fig_p027_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Total number of subscribers as a function of [PITH_FULL_IMAGE:figures/full_fig_p028_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: NO’s profit Π∗ 1 as a function of 1 − γ for different values of δ (strategic) 28 [PITH_FULL_IMAGE:figures/full_fig_p028_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: VO’s profit Π∗ 2 as a function of γ for different values of δ (strategic) 4.3. Business model feasibility In this section, the conditions are discussed under which the business models proposed in Section 2 are feasible, i.e., provide incentives to both the NO and the VO. First, the monopolistic business model is tackled, and then, the strategic one. 4.3.1. Monopolistic business model The conditions are se… view at source ↗
Figure 30
Figure 30. Figure 30: Π∗ 1 + Π∗ 2 and Π∗m as a function of γ for α1 = 0.6 and α2 = 0.4 and different values of δ [PITH_FULL_IMAGE:figures/full_fig_p031_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Π∗ 1 + Π∗ 2 and Π∗m as a function of γ for α1 = 0.6 and α2 = 0.6 and different values of δ 31 [PITH_FULL_IMAGE:figures/full_fig_p031_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Π∗ 1 + Π∗ 2 and Π∗m as a function of γ for α1 = 0.6 and α2 = 0.8 and different values of δ in the three figures and the VO’s user base has sensitivities α2 equal to 0.4 ( [PITH_FULL_IMAGE:figures/full_fig_p032_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: n ∗ 1 + n ∗ 2 in the strategic and the monopolistic scenarios as a fuction of γ for α1 = 0.6 and α2 = 0.4 and different values of δ [PITH_FULL_IMAGE:figures/full_fig_p033_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: n ∗ 1 + n ∗ 2 in the strategic and the monopolistic scenarios as a fuction of γ for α1 = 0.6 and α2 = 0.6 and different values of δ 33 [PITH_FULL_IMAGE:figures/full_fig_p033_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: n ∗ 1 + n ∗ 2 in the strategic and the monopolistic scenarios as a fuction of γ for α1 = 0.6 and α2 = 0.8 and different values of δ had different sensitivities. The utility expression for all users has been the same. An improved appraisal of the user heterogeneity could be obtained by modeling each user base with a different utility expression. And thirdly, an issue that receives frequent attention from t… view at source ↗
read the original abstract

The provision of services by more than one operator over a common network infrastructure, as enabled by 5G network slicing, is analyzed. Two business models to be implemented by a network operator, who owns the network, and a virtual operator, who does not, are proposed. In one business model, named \emph{strategic}, the network operator provides service to its user base and the virtual operator provides service to its user base and pays a per-subscriber fee to the network operator. In the other business model, named \emph{monopolistic}, the network operator provides service to both user bases. The two proposals are analyzed by means of a model that captures both system and economic features. As regards the systems features, the slicing of the network is modeled by means of a Discriminatory Processor Sharing queue. As regards the economic features, the incentives are modeled by means of the user utilities and the operators' revenues; and game theory is used to model the strategic interaction between the users' subscription decision and the operators' pricing decision. In both business models, it is shown that the network operator can be provided with the appropriate economic incentives so that it acquiesces in serving the virtual operator's user base (monopolistic model) and in allowing the virtual operator to provide service over the network operator's infrastructure (strategic model). From the point of view of the users, the strategic model results in a higher subscription rate than the monopolistic model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes two business models for a network operator and virtual operator in 5G slicing: a strategic model in which the virtual operator pays a per-subscriber fee while serving its own users, and a monopolistic model in which the network operator serves both user bases. System performance is captured via a discriminatory processor sharing (DPS) queue, while economic incentives are modeled through user utilities linear in price and delay together with a two-stage game whose equilibria determine subscription and pricing decisions. The central claims are that the network operator obtains positive revenue gain (hence appropriate incentives) from serving the virtual operator’s traffic in both models, and that the strategic model produces a strictly higher equilibrium subscription rate.

Significance. If the closed-form equilibria hold, the work supplies a concrete, parameter-light demonstration that pricing can align incentives for virtual operators on shared 5G infrastructure. The integration of exact DPS mean-sojourn formulas with a two-stage game yields falsifiable conditions on fees and service rates, which is a methodological strength. The result is relevant to network economics and 5G business-model design, provided the queueing and utility primitives remain representative.

major comments (2)
  1. [§4.2] §4.2, two-stage game equilibrium (around Eq. (14)–(17)): the network operator’s best response includes the virtual operator’s traffic only because the first-order condition on revenue is positive under the linear utility assumption. Replacing the linear demand with a concave form (e.g., log utility) reverses the sign of that derivative for the same parameter region, eliminating the claimed incentive compatibility. A robustness check with non-linear demand is required.
  2. [§3.1] §3.1, DPS mean-delay formulas (Eq. (5)–(7)): the model assumes zero isolation or orchestration overhead. Introducing even a modest multiplicative reduction (e.g., 10 %) to each slice’s effective service rate alters the mean sojourn times and can flip the sign of the network operator’s revenue gain in both the monopolistic and strategic equilibria. The headline incentive-alignment result is therefore sensitive to this unmodeled factor.
minor comments (2)
  1. [§2] Notation for the two user classes (NO vs. VO) is introduced inconsistently between §2 and §4; a single table of symbols would improve readability.
  2. [Figure 3] Figure 3 caption does not list the exact parameter values used for the plotted equilibria, making reproduction difficult.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important sensitivities in our modeling assumptions. We address each point below and will revise the manuscript accordingly to strengthen the analysis.

read point-by-point responses

  1. Referee: [§4.2] §4.2, two-stage game equilibrium (around Eq. (14)–(17)): the network operator’s best response includes the virtual operator’s traffic only because the first-order condition on revenue is positive under the linear utility assumption. Replacing the linear demand with a concave form (e.g., log utility) reverses the sign of that derivative for the same parameter region, eliminating the claimed incentive compatibility. A robustness check with non-linear demand is required.

    Authors: We acknowledge that the positive revenue incentive in the strategic model relies on the linear utility form, which yields a closed-form equilibrium but may not hold under concave utilities. Linear utilities are standard in this literature for tractability when modeling subscription decisions based on price and delay. In the revised version we will add a robustness subsection deriving the equilibrium conditions under logarithmic utility, identifying the parameter ranges (e.g., on service rates and fees) where the network operator’s incentive to serve the virtual operator remains positive, and noting where the result is sensitive. revision: yes

  2. Referee: [§3.1] §3.1, DPS mean-delay formulas (Eq. (5)–(7)): the model assumes zero isolation or orchestration overhead. Introducing even a modest multiplicative reduction (e.g., 10 %) to each slice’s effective service rate alters the mean sojourn times and can flip the sign of the network operator’s revenue gain in both the monopolistic and strategic equilibria. The headline incentive-alignment result is therefore sensitive to this unmodeled factor.

    Authors: The DPS analysis assumes ideal slicing with no orchestration overhead, consistent with the focus on economic incentives under perfect resource sharing. We agree this is a simplification that could affect sojourn times and revenue signs in practice. The revised manuscript will include a sensitivity analysis introducing a multiplicative overhead factor (0.9–1.0) to the slice service rates, recomputing the mean delays and equilibrium revenues, and reporting the overhead thresholds below which the network operator’s revenue gain remains positive in both models. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no circular reductions to inputs or self-citations

full rationale

The paper models network slicing via the standard discriminatory processor-sharing (DPS) queue and analyzes incentives via a two-stage game with linear user utilities in price and delay. The claimed incentive compatibility results are obtained by solving the equilibrium conditions of this model for parameter ranges that yield positive revenue for the network operator. No equation reduces by construction to a fitted parameter renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The DPS mean-sojourn formulas and game payoffs are derived from first principles within the paper and remain falsifiable against external benchmarks (realistic overheads or non-linear demand). The central claim therefore rests on independent model content rather than tautological re-expression of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on two domain assumptions about modeling choices and no identifiable free parameters or new entities.

axioms (2)
  • domain assumption Network slicing performance is accurately captured by a discriminatory processor sharing queue.
    Explicitly stated as the system model in the abstract.
  • domain assumption User subscription decisions and operator pricing interactions are captured by game theory using utilities and revenues.
    Explicitly stated as the economic modeling approach in the abstract.

pith-pipeline@v0.9.0 · 5575 in / 1406 out tokens · 42205 ms · 2026-05-16T11:57:09.817233+00:00 · methodology

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    NGMN Alliance, 5G white paper, Next Generation Mobile Networks, White paper (2015)

    work page 2015

  2. [2]

    Naor, The regulation of queue size by levying tolls, Econometrica: journal of the Econometric Society (1969) 15–24

    P. Naor, The regulation of queue size by levying tolls, Econometrica: journal of the Econometric Society (1969) 15–24. 34

    work page 1969

  3. [3]

    Allon, A

    G. Allon, A. Federgruen, Service competition with general queueing facilities, Operations Research 56 (4) (2008) 827–849

    work page 2008

  4. [4]

    Hassin, M

    R. Hassin, M. Haviv, To queue or not to queue: Equilibrium behavior in queueing systems, Vol. 59, Springer Science & Business Media, 2003

    work page 2003

  5. [5]

    Hassin, Rational queueing, Chapman and Hall/CRC, 2016

    R. Hassin, Rational queueing, Chapman and Hall/CRC, 2016

    work page 2016

  6. [6]

    Mandjes, Pricing strategies under heterogeneous service requirements, Computer Networks 42 (2) (2003) 231–249

    M. Mandjes, Pricing strategies under heterogeneous service requirements, Computer Networks 42 (2) (2003) 231–249

    work page 2003

  7. [7]

    Hayel, B

    Y. Hayel, B. Tuffin, Pricing for heterogeneous services at a discriminatory processor sharing queue, in: International Conference on Research in Networking, Springer, 2005, pp. 816–827

    work page 2005

  8. [8]

    Guijarro, V

    L. Guijarro, V. Pla, B. Tuffin, Entry game under opportunistic access in cognitive radio networks: A priority queue model, in: 2013 IFIP Wireless Days (WD), IEEE, 2013, pp. 1–6

    work page 2013

  9. [9]

    Sanchis-Cano, L

    A. Sanchis-Cano, L. Guijarro, V. Pla, J. R. Vidal, Economic viability of htc and mtc service provi- sion on a common network infrastructure, in: 2017 14th IEEE Annual Consumer Communications & Networking Conference (CCNC), IEEE, 2017, pp. 1044–1050

    work page 2017

  10. [10]

    E. J. Sacoto-Cabrera, A. Sanchis-Cano, L. Guijarro, J. R. Vidal, V. Pla, Strategic interaction between operators in the context of spectrum sharing for 5G networks, Wireless Communications and Mobile Computing 2018 (2018)

    work page 2018

  11. [11]

    Hayel, D

    Y. Hayel, D. Ros, B. Tuffin, Less-than-best-effort services: Pricing and scheduling, in: IEEE INFOCOM 2004, Vol. 1, IEEE, 2004

    work page 2004

  12. [12]

    Reichl, B

    P. Reichl, B. Tuffin, R. Schatz, Logarithmic laws in service quality perception: where microeconomics meets psychophysics and quality of experience, Telecommunication Systems (2011) 1–14

    work page 2011

  13. [13]

    Maschler, E

    M. Maschler, E. Solan, S. Zamir, Game Theory, Cambridge University Press, 2013

    work page 2013

  14. [14]

    J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the institution of civil engineers 1 (3) (1952) 325–362. 35

    work page 1952