A Repairable System Supported by Two Spare Units and Serviced by Two Types of Repairers

Jyotirmoy Sarkar; Vahid Andalib

REVIEW 3 major objections 4 minor 16 references

A Repairable System Supported by Two Spare Units and Serviced by Two Types of Repairers

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For all four repair policies, the long-run availability and profit of a two-spare repairable system are derived exactly from a six-state semi-Markov process.

desk verdict The two-spare random-patience models are a solid but modest extension; the deterministic-patience models have a load-bearing Markovianity flaw that sinks the paper's headline recommendation. read the letter →

arxiv 1908.02547 v1 pith:G2M44ZEM submitted 2019-08-04 cs.PF cs.SYeess.SY

Vahid Andalib , Jyotirmoy Sarkar This is my paper · ORCID

classification cs.PFcs.SYeess.SY MSC 60K1560K2090B25

keywords coldstandbyrepairablesystemsemi-Markovprocesslimitingavailabilityprofitperunittimepatiencemultiplerepairbyexpertsingle

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a single operating unit backed by two identical cold-standby spares. When the operating unit fails, the regular repairer works on it until either repair completes, a patience time expires, or the system goes down; if the patience time expires or the system fails, a faster but more expensive expert repairer is called. The central claim is that for all four policies—single versus multiple expert repairs, and random versus deterministic patience time—the long-run fraction of time the system is up and the long-run profit per unit time are given exactly by the stationary formulas of a six-state semi-Markov process under exponential lifetimes and repair times. If the claim is right, a maintenance engineer can compute the best patience time and expert workload from the model parameters, and the numerical comparisons show that repairing all failed units beats repairing one per visit on availability while a suitably tuned deterministic patience time can beat a random one on profit.

What carries the argument

The central object is the six-state semi-Markov process, a process that moves between states by a Markov chain after random sojourn times, with states recording how many units are operating, on standby, under regular repair, under expert repair, or awaiting repair. The load-bearing identity is $\theta_k = \pi_k \mu_k / \sum_{j=1}^6 \pi_j \mu_j$, which converts the embedded chain's stationary probabilities and mean sojourn times into long-run occupancy proportions; the paper then computes availability as $A_\infty = 1 - \theta_6$ and profit per unit time as $\omega = A_\infty(R_p-C_p) - [\Theta_r C_r + \Theta_e C_e + C_l/\tau]$, with $\tau$ obtained from recursive expected-cycle equations.

What would settle it

Compare the paper's $P_{45}$ with the value obtained by conditioning on the actual event that the system enters State 4, namely $X < \min(Y,T)$, rather than only on $X < T$; a discrepancy in the transition probability, or a mismatch between simulated $\theta_6$ and equations (4.20) or (4.23), would show that the deterministic-patience models are approximations.

Watch

Extended reading notes

Core claim

Under exponential life and repair times (lifetime rate $\lambda$, regular repair rate $\beta$, expert repair rate $\gamma$, patience rate $\alpha$), the system is modeled as a semi-Markov process whose embedded chain has six states recording how many units are operating, on standby, under regular repair, under expert repair, or awaiting repair. The paper derives the transition matrix $P$, the stationary distribution $\pi$ of the embedded chain, and the mean sojourn times $\mu_k$ for each of the four models, and applies $\theta_k = \pi_k \mu_k / \sum_j \pi_j \mu_j$. This yields $A_\infty = 1 - \theta_6$ and the profit rate $\omega = A_\infty(R_p-C_p) - (\Theta_r C_r + \Theta_e C_e + C_l/\tau)$, where $\tau$ is the expected cycle time obtained by solving recursive relations and $C_l/\tau$ follows from Wald's identity. The paper concludes that multiple-repair-by-expert dominates single-repair-by-expert in availability, that adding a second spare raises both availability and profit, and that for any cost parameters there is a threshold expert rate separating when MRE or SRE is more profitable, along with an optimal patience time under the deterministic-patience policy.

Load-bearing premise

The exact formulas for the deterministic-patience models assume the six-state embedded chain remains Markovian after part of the patience time has already elapsed; if the remaining patience depends on that elapsed time in a way the state does not record, Models 3 and 4 are approximations rather than exact.

Editorial extensions

If this is right

The paper concludes that MRE yields higher limiting availability than SRE for both random and deterministic patience times, so if uptime is the priority and the expert can be afforded, the expert should repair all failed units during a visit.
A system with two cold-standby spares has higher limiting availability and limiting profit than the same system with one spare across all four policies, which quantifies the value of the extra spare when availability falls below an acceptable threshold.
For any parameter choice there is a patience-time interval where the deterministic-patience policy dominates the random-patience policy in profit, and within the deterministic policy there is an optimal patience time $T$ maximizing $\omega$, so an engineer who prefers the logistically simpler deterministic rule can use that interval.
There is a threshold on the expert's cost rate: below it MRE is more profitable than SRE, above it SRE is more profitable, and knowing this threshold decides how many failed units the expert should repair per visit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension not pursued in the paper is to re-derive the deterministic-patience models with an augmented state recording the remaining patience time; comparing those formulas with the paper's closed forms would show how much accuracy the six-state approximation sacrifices.
The same semi-Markov setup should extend to three or more spares by adding states for queues of failed units, with availability approaching one but profit eventually bounded by repair and trip costs.
A concrete numerical check would simulate the four models at the paper's example parameters to confirm that MRE-DPT maximizes both $A_\infty$ and $\omega$ in the patience interval $[1.45, 1.62]$ and that $\omega$ peaks near $T=2.19$; this would also expose any bias introduced by the deterministic-patience Markov assumption.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Quick take: the paper is a genuine but modest extension of Bieth-Hong-Sarkar's one-spare model to two spares, and the RPT half is mostly careful work. The DPT half is wrong in a load-bearing way, and the paper's own conclusion—that a tuned deterministic patience time beats exponential patience—rests on that half.

What's new and good: the state space grows from the one-spare model, and the recursive cycle-time equations adapt cleanly to MRE and SRE under exponential patience. The derivations in Sections 4.1 and 4.2 look coherent; I spot-checked the stationary structure and it is consistent. The comparison figures also make sense for the RPT models. Credit where due: it is clearly written and the algebra is checked.

The soft spot is not minor. Models 3 and 4 assume the embedded six-state chain is Markovian when the patience time is deterministic, but the residual patience T' = T - X depends on the lifetime X that just occurred. That history is not in the state. So the transition probabilities out of State 4 depend on the age of the regular repair, and Theorem 4.1 does not apply. The P45 derivation in Section 4.3 confirms the problem: it averages the residual patience over the unconditional density of X, rather than conditioning on X < min(Y,T), the event that actually puts the system in State 4. Correcting that conditioning gives a different formula, and even then the state is still not sufficient. The paper tips its hand in Section 6(ii), where it says the Markovian property fails under DPT—that is true for the current model, not just the future two-facility version.

So the RPT models stand, the DPT models fall, and the central practical claim ("MRE-DPT is best for T in [1.45,1.62]") is unsupported. The paper could be repaired by dropping the DPT models or reworking them with an augmented state or a regenerative analysis, but as submitted it needs major revision.

Who this is for: reliability engineers who want the two-spare RPT formulas, and readers who want a cautionary example of why residual times must be in the state. It deserves a serious referee—the RPT half is sound and worth having—but I would reject the current version. Send it to peer review only if the editor expects heavy revision; otherwise desk reject and invite a resubmission with DPT fixed.

Referee Report

3 major / 4 minor

Summary. The manuscript extends the one-spare-unit system of Bieth, Hong, and Sarkar [2] to a one-unit system supported by two cold-standby spares, serviced by a regular in-house repairer and a visiting expert repairer. Four policies are considered, formed by crossing single versus multiple repair by the expert (SRE versus MRE) with random versus deterministic patience time (RPT versus DPT). The authors model the system as a six-state semi-Markov process (SMP), derive the stationary state proportions in closed form, and use them to compute the limiting availability A_inf and limiting profit per unit time omega for each model. The main advertised conclusions are that MRE dominates SRE in availability, that for suitable deterministic patience time the DPT policy dominates the RPT policy in profit, and that for the numerical example the MRE-DPT model is optimal for patience times in [1.45,1.62].

Significance. If the derivations were correct, the paper would provide a useful extension of [2] and a concrete maintenance-engineering decision rule. The RPT models (Sections 4.1 and 4.2) are standard SMP calculations and appear internally coherent; the extension from one to two spare units is natural, and the explicit algebraic formulas are a strength. However, the DPT models (Sections 4.3 and 4.4) are not valid SMPs: the residual patience time is not Markovian in the six-state chain, and the transition probabilities out of State 4 are derived using an incorrect conditioning. Since the central profit comparison, including the claimed optimal DPT range, depends on these models, the main new conclusions are unsupported. The paper also makes useful numerically illustrated comparisons, but no code is provided.

major comments (3)

[Section 4, Theorem 4.1 and Sections 4.3-4.4] The justification that the process is an SMP applies only to the RPT models. Under a deterministic patience time T, the remaining patience when the system enters State 4 is T' = T - X, where X is the already-realized lifetime in State 2; this quantity is not exponentially distributed and is not part of the six-state description. Consequently the transition probabilities out of State 4 depend on the history X, the embedded six-state chain is not a Markov chain, and Theorem 4.1 cannot be applied to Models 3 and 4. The paper itself notes in Section 6(ii) that the Markovian property fails under the DPT policy in a related setting. Because the formulas (4.18)-(4.23) and the Section 5 conclusions, including the optimal range [1.45,1.62], rest on this application, the DPT-based central claims are not supported.
[Section 4.3, Eq. (4.18), derivation of P45] The derivation of P45 conditions on the wrong event and uses the wrong density. State 4 is entered from State 2 when X < min(Y,T), not merely when X < T, and the conditional density of X given this entry event is (lambda+beta)e^{-(lambda+beta)x}/(1-e^{-(lambda+beta)T}) for x in [0,T], not the unconditional density lambda e^{-lambda x}. The displayed integral has no conditioning denominator and therefore yields the incorrect value lambda e^{-lambda T}(1-e^{-beta T})/beta. Conditioning correctly gives P45 = (lambda+beta)T e^{-(lambda+beta)T}/(1-e^{-(lambda+beta)T}). Since P42 and P46 are then defined from this P45, the transition probabilities and the stationary distribution (4.19) are incorrect.
[Section 4.4, Eq. (4.21), SRE-DPT model] In the SRE model the transition matrix sends State 6 to State 4, so State 4 is reached not only from State 2 (with residual patience T-X) but also after an expert visit. The paper simply reuses the MRE-DPT values of P42, P45, and P46 in equation (4.21). These two entry paths have different patience-time clocks unless the model explicitly specifies that patience restarts and unless the state label encodes the entry type. Without such an augmentation, the transition probabilities out of State 4 are path-dependent, and the stationary distribution (4.22) together with the SRE-DPT values of A_inf and omega are unsupported.

minor comments (4)

[Title page / affiliation] The affiliation line contains a typo: "Indiana Universiry-Purdue University Indianapolis" should read "Indiana University-Purdue University Indianapolis."
[Section 3, Eq. (3.2)] The statement that Wald's First Identity gives the expected number of expert visits per unit time as the reciprocal of the cycle length is imprecise; the renewal reward theorem is the standard result for the long-run rate of renewals, and citing it would be more accurate.
[Section 5, item after Figure 4] There is a typo: "th opposite" should be "the opposite."
[Sections 5, Figures 2 and 3] The numerical comparisons are presented for a single parameter set without supporting code or tabulated values; since the paper's main claim is a comparison across policies, it would be helpful to state explicitly whether the figures are generated from the displayed closed forms or by simulation, and ideally to provide the data or code used.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the central two-spare SMP derivations are self-contained, with only a minor non-load-bearing self-citation.

full rationale

The paper's central claims — the limiting availability A_inf and limiting profit omega for the four two-spare models — are obtained algebraically from explicitly stated transition matrices, sojourn-time expectations, and the standard SMP formula in Theorem 4.1. No fitted parameter is later renamed as a prediction, and no target quantity is used as an input to its own derivation. The only self-citation is reference [2], by Bieth, Hong, and Sarkar (Sarkar is a co-author of this paper), which is used as a one-spare-unit baseline and for the one-spare crossover time T*; it is not load-bearing for the new two-spare formulas. The paper is self-contained in deriving the two-spare results, and its comparisons with the one-spare system are external benchmarks rather than circular inputs. A genuine concern exists that the DPT models are not semi-Markov because the remaining patience time T' = T - X after entering State 4 depends on the history, and the paper itself notes in Section 6(ii) that 'the Markovian property fails under the DPT policy' in a related setting; however, that is a mathematical validity issue, not a circularity pattern under the specified definitions. Accordingly, the circularity score is low.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameters are fitted to data; all rates and costs are inputs to the model and the numerical illustrations. The main unstated load is the semi-Markov assumption for the DPT models, which is not justified because the residual deterministic patience time is history-dependent.

assumptions (4)

domain assumption Life times, repair times, and patience times are exponentially distributed and mutually independent.
Assumption 8 in Section 3. This makes the RPT models semi-Markov and gives the closed-form sojourn times in (4.2).
standard math The embedded six-state discrete-time chain is irreducible with finite mean and non-lattice recurrence times, so the SMP limiting-probability theorem applies.
Invoked before Theorem 4.1 in Section 4. The paper does not verify irreducibility or the non-lattice condition for each of the four models.
ad hoc to paper For deterministic patience, the process remains an SMP with the same six-state chain even though the residual patience T-X is not part of the state.
Used in Sections 4.3 and 4.4. This premise is fragile because the transition out of State 4 depends on the elapsed lifetime X, and the paper itself notes in Section 6 that the Markovian property fails under DPT in a related setting.
standard math Each cycle contains exactly one expert visit, so Wald's identity gives the trip charge per unit time as Cl divided by the expected cycle length.
Used in equation (3.2) to express the limiting profit per unit time.

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Pith. "Pith review of A Repairable System Supported by Two Spare Units and Serviced by Two Types of Repairers." pith.science (2026). https://pith.science/paper/G2M44ZEM

@misc{pith2026190802547,
  author       = {Pith},
  title        = {Pith review of: A Repairable System Supported by Two Spare Units and Serviced by Two Types of Repairers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2M44ZEM}},
  note         = {Machine review of arXiv:1908.02547}
}

abstract

We study a one-unit repairable system, supported by two identical spare units on cold standby, and serviced by two types of repairers. The model applies, for instance, to ANSI (American National Standard Institute) centrifugal pumps in a chemical plant. The failed unit undergoes repair either by an in-house repairer within a random or deterministic patience time, or else by a visiting expert repairer. The expert repairs one or all failed units before leaving, and does so faster but at a higher cost rate than the regular repairer. Four models arise depending on the number of repairs done by the expert and the nature of the patience time. We compare these models based on the limiting availability $A_{\infty}$, and the limiting profit per unit time $\omega$, using semi-Markov processes, when all distributions are exponential. As anticipated, to maximize $A_{\infty}$, the expert should repair all failed units. To maximize $\omega$, a suitably chosen deterministic patience time is better than a random patience time. Furthermore, given all cost parameters, we determine the optimum number of repairs the expert should complete, and the optimum patience time given to the regular repairer in order to maximize $\omega$.

Figures

Figures reproduced from arXiv: 1908.02547 by the authors.

**Figure 2.** Limiting availability under all repair models for systems with one spare unit (dashed) and two spare units [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Limiting profit per unit time under all repair models for the systems with one spare unit (dashed) and two [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Limiting profit per unit time as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 13 canonical work pages

[2]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

A standby system with two types of repair persons

Bruno Bieth, Liang Hong, and Jyotirmoy Sarkar. A standby system with two types of repair persons. Applied Stochastic Models in Business and Industry, 26(5):577–594, 2010. https://doi.org/10.1002/asmb.801

work page doi:10.1002/asmb.801 2010

[2] [1]

An introduction to probability theory and its applications, volume 1

William Feller. An introduction to probability theory and its applications, volume 1. Wiley, New York, 1968

work page 1968

[3] [3]

Limiting average availability of a system supported by several spares and several repair facilities

Jyotirmoy Sarkar and Fang Li. Limiting average availability of a system supported by several spares and several repair facilities. Statistics & probability letters, 76(18):1965–1974, 2006. https://doi.org/10.1016/j.spl. 2006.04.046. 12 A Repairable System Supported by Two Spare Units and Serviced by Two Types of RepairersA PREPRINT

doi:10.1016/j.spl 1965

[4] [4]

Availability of a one-unit system supported by several spares and repair facilities

Jyotirmoy Sarkar and Atanu Biswas. Availability of a one-unit system supported by several spares and repair facilities. Journal of the Korean Statistical Society, 39(2):165–176, 2010. https://doi.org/10.1016/j.jkss. 2009.05.001

doi:10.1016/j.jkss 2010

[5] [5]

Reliability and sensitivity analysis of a repairable system with warm standbys and r unreliable service stations

Kuo-Hsiung Wang, Jyh-Bin Ke, and Wen-Chiung Lee. Reliability and sensitivity analysis of a repairable system with warm standbys and r unreliable service stations. The International Journal of Advanced Manufacturing Technology, 31(11-12):1223–1232, 2007. https://doi.org/10.1007/s00170-005-0298-0

work page doi:10.1007/s00170-005-0298-0 2007

[6] [6]

A deteriorating cold standby repairable system with priority in use.European Journal of Operational Research, 183(1):278–295, 2007

Yuan Lin Zhang and Guan Jun Wang. A deteriorating cold standby repairable system with priority in use.European Journal of Operational Research, 183(1):278–295, 2007. https://doi.org/10.1016/j.ejor.2006.09.075

work page doi:10.1016/j.ejor.2006.09.075 2007

[7] [7]

Optimal design of a maintainable cold-standby system

Haiyang Yu, Farouk Yalaoui, ˙Eric Châtelet, and Chengbin Chu. Optimal design of a maintainable cold-standby system. Reliability Engineering & System Safety, 92(1):85–91, 2007. https://doi.org/10.1016/j.ress. 2005.11.001

doi:10.1016/j.ress 2007

[8] [8]

Stochastic analysis of a two-unit cold standby system with two-stage repair and waiting time

Khaled M El-Said and Mohamed S El-Sherbeny. Stochastic analysis of a two-unit cold standby system with two-stage repair and waiting time. Sankhya B , 72(1):1–10, 2010. https://doi.org/10.1007/ s13571-010-0001-9

work page 2010