Mitigating long queues and waiting times with service resetting

Ofek Lauber Bonomo1, Arnab Pal2 and Shlomi Reuveni11School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences & The Center for

Physics and Chemistry of Living Systems & The Ratner Center for Single Molecule Science,Tel Aviv University, Tel Aviv 6997801, Israel and

2Department of Physics, Indian Institute of Technology, Kanpur, Kanpur 208016, India(Dated: November 4, 2021)

What determines the average length of a queue which stretches in front of a service station? Theanswer to this question clearly depends on the average rate at which jobs arrive to the queue andon the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations inservice and arrival times are also important, and that these are a major source of backlogs anddelays. Strategies that could mitigate fluctuations induced delays are in high demand as queuestructures appear in various natural and man-made systems. Here we demonstrate that a simpleservice resetting mechanism can reverse the deleterious effects of large fluctuations in service times,thus turning a marked drawback into a favourable advantage. This happens when stochastic fluc-tuations are intrinsic to the server, and we show that the added feature of service resetting canthen dramatically cut down average queue lengths and waiting times. While the analysis presentedherein is based on the M/G/1 queueing model where service is general but arrivals are assumedto be Markovian, Kingman’s formula asserts that the benefits of service resetting will carry overto queues with general arrivals. We thus expect results coming from this work to find widespreadapplication to queueing systems ranging from telecommunications, via computing, and all the wayto molecular queues that emerge in enzymatic and metabolic cycles of living organisms.

I. INTRODUCTION

Queueing theory is the mathematical study of waitinglines [1–4]. Ranging from the all familiar supermarketand bank, to call centers [5, 6], airplane boarding [7–9],telecommunication and computer systems [10–15], pro-duction lines and manufacturing [16–19], enzymatic andmetabolic pathways [20–27], gene expression [28–34], andin transport phenomena [35–46], waiting lines and queuesappear ubiquitously and play a central role in our lives.While the teller at the bank works at a (roughly) con-stant rate, other servers, e.g., computer systems [14], andmolecular machines like enzymes [47–52], often displaymore pronounced fluctuations in service times. Thesefluctuations have a significant effect on queue perfor-mance [1–4]: higher fluctuations in service times will re-sult in longer queues as illustrated in Fig. 1a. Servicetime variability is thus a major source of backlogs anddelays in queues [14], and this problem is particularlyacute when encountering heavy tailed workloads [53].

Different strategies have been developed to mitigatethe detrimental effect caused by stochastic service timefluctuations. In particular, when considering singleserver queues, various scheduling policies can be appliedto reduce waiting times. For example, computer serverscan be designed to serve smaller jobs first, rather than byorder of arrival. While this policy can be criticized fromthe standpoint of fairness, it does reduce average waitingtimes by preventing situations where typically sized jobs(which are common) get stuck behind a single job that isvery large [14]. In situations where service can be stoppedand continued from the same point at a later time, onecan further improve performance by implementing a pol-icy in which only the job with the shortest remaining

𝝀

ServiceTime

(a) (b)

ServiceTime

𝝀 𝝀 𝝀

FIG. 1. Backlogs and delays caused by service time fluctua-tions can be mitigated with service resetting. Panel (a): Themean number of agents—jobs, customers, molecules, etc.—in a queue is sensitive to stochastic fluctuations in the servicetime provided by the server (computer, cashier, enzyme, etc.).For a given arrival rate and mean service time, the queuestretching in front of the server whose service time distribu-tion is more variable will be longer on average, as illustratedby the blue and red servers in the figure. Panel (b): We showthat when large service time fluctuations are intrinsic to theserver, resetting the service process can lower both the meanand variance of the service time and drastically improve queueperformance.

service time is served [54]. This policy can be provenoptimal under certain conditions [57].

The problem with the above-mentioned schedulingpolicies is that they are ill-equipped to deal with situ-ations where fluctuations in service times are intrinsic tothe serves itself. These are in fact prevalent. For exam-ple, stochastic optimization algorithms can take wildly

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different times to solve two instances of the exact sameproblem [58]. Similarly, the time it takes an enzymeto catalyze a given chemical reaction varies considerablybetween turnover cycles — despite the fact that the in-coming (substrate) and outgoing (product) molecules arechemically identical [47–52]. Crucially, in these and sim-ilar cases the total service time of an incoming job isunknown a priori and the time remaining for a job inservice cannot be determined. Sized-based schedulingpolicies are thus impossible to implement, which callsfor the development of new and novel approaches to theproblem.

In this paper we propose a complementary approach tosolve the problems caused by service time fluctuations inqueueing systems. Namely, we show that implementing asimple service resetting policy can reverse the deleteriouseffect of stochastic fluctuations when the latter are largeand particularly harmful. The policy, which consists ofrandom or deterministic resetting of the service process,may seem counterintuitive at first. After all, no bene-fit can possibly come from doing the exact same thingall over again. However, here we show that when ser-vice time fluctuations are intrinsic to the server itself, inthe sense that jobs whose service has been reset are as-signed fresh service times, service resetting can be usedto drastically improve queue performance as illustratedin Fig. 1b. Indeed, it has recently been shown that reset-ting has the ability to expedite the completion of randomprocesses: from stochastic optimization [60–63], via first-passage and search [64–80], and onto chemical reactions[81–83], and this general principle is hereby exploited inthe context of queueing.

A different way in which resetting can be used to af-fect queue performance is by resetting of the jobs arrivalprocess, which is interesting to consider e.g., in the con-text of intracellular transport [84]. However, resettingthe arrival process has no affect on service time fluctu-ations, which is further compounded by the fact that inmost conventional queue structures the arrival process isextrinsic to the system and is thus not subject to controlor optimization. We thus focus on queues with serviceresetting to which we devote this paper. Before mov-ing forward to develop the theory and discuss examples,we mention in passing that the concept of resetting isalso relevant in the context of queues with catastrophicevents [85–99]. Note, however, that such catastrophicevents lead to mass annihilation of jobs (agents) from thequeue. Contrary, service resetting conserves the numberof jobs in the queue, as jobs whose service was reset wouldstill require full service before they can leave the system.Thus, ‘resetting by catastrophe’ should not to be con-fused with the service resetting policy that is introducedand rigorously analyzed below.

The remainder of this paper is structured as follows. InSec. II, we provide a brief review of the M/G/1 queuingmodel for which queue arrivals are Markovian and serviceis general. This queue will serve as the main modelingplatform of this paper. We also review the Pollaczek-

Khinchin formula which provides the mean number ofjobs in the M/G/1 queue at the steady-state, highlight-ing the sensitivity of the latter to service time fluctua-tions. Finally we provide a short overview of existingscheduling strategies aimed to mitigate service time fluc-tuations. In Sec. III, we discuss the service resettingpolicy and formulate the M/G/1 model with service re-setting. We show the latter can be mapped onto thestandard M/G/1 queue, but with a modified service timedistribution that depends on the underlying distributionsof the service and resetting times. This fact allows us tore-apply the Pollaczek-Khinchin formula to study howthe mean number of jobs in the queue depends on theresetting protocol. We focus on two prominent resettingprotocols for which we provide detailed analysis: In Sec.IV we consider Poissonian resetting. i.e., resetting at aconstant rate. We show that when the variability in theunderlying service time distribution is high, the meannumber of jobs in the system obtains a minimum as afunction of the resetting rate. This means that resettingcan drastically improve queue performance. Similarly, inSec. V we study resetting at fixed time intervals (a.k.asharp resetting), and show that while this resetting pro-tocol leads to similar qualitative results, the fixed inter-resetting time duration can be tuned to perform betterthan any other stochastic resetting protocol (Poissonianresetting included). In Sec. VI we illustrate the gen-eral results obtained for two well-established service timedistributions, namely the log-normal and Pareto distri-bution, which are considered respectively in subsectionsVI A and VI B. We end in Sec. VII with conclusions andoutlook.

In what follows we use fZ(t), qZ(t) = 1−∫ t0dτ fZ(τ),

〈Z〉, Var(Z), and Z(s) ≡ 〈e−sZ〉 to denote, respectively,the probability density function, the survival function,expectation, variance, and Laplace transform of a non-negative random variable Z.

II. QUEUES: MODELS AND PRELIMINARIES

Consider a queuing system that is composed of a queuein which jobs await to be served, and a server whichserves one job at a time according to a First-Come, First-Served policy (FCFS). A queue can be identified as astochastic process whose state space is denoted by theset N = {0, 1, 2, 3, ...}, where the value corresponds tothe number of jobs in the queue, including the one be-ing served. In particular, an M/G/1 queue (written inKendall’s notation [100]) consists of a single server andqueue to which jobs arrive according to a Poisson processwith rate λ [13, 14]. Thus, M stands for Markovian ormemory-less arrival process. No assumptions are madeon the service time of jobs, which comes from a generaldistribution (hence the G in Kendall’s notation). De-noting the service time by the random variable S (forservice), whose density we denote by fS(t), we define theservice rate µ ≡ 1/〈S〉. It will prove convenient to intro-

3

duce the utilization parameter ρ = λ/µ, which gives thefraction of time the server is working (not idle) in thesteady state [14]. The model attains a steady state aslong as the arrival rate is smaller than the service rate,i.e, ρ < 1. For ρ > 1 the queue grows indefinitely longwith time and the system does not attain a steady state.Thus, in what follows we will assume ρ < 1.

The number of jobs in an M/G/1 queue fluctuates withtime. Thus, it is only natural to start the discussion withthe average of this observable. The mean number of jobs〈N〉 in the system (queue+server) is given by the famousPollaczek-Khinchin formula [14]

〈N〉 =ρ

1− ρ+

ρ2

2(1− ρ)

(CV 2 − 1

), (1)

where ρ is the utilization, and CV 2 = Var(S)/〈S〉2 isthe squared coefficient of variation or the variability inservice time. As expected, the mean number of jobs is amonotonically increasing function of the utilization, with〈N〉 tending to infinity as ρ → 1−. Thus, as expected,low service rates lead to long queues. Note, however, theappearance of CV 2 in the second term of Eq. (1). Thisindicates that the mean length of the queue is highlysensitive to stochastic fluctuations in the service time,which is a non-trivial effect that can be explained by theinspection paradox [14].

From the second term in Eq. (1), we see that the meannumber of jobs is a monotonically increasing functionof CV 2. Note, that when CV < 1, i.e., when servicetime fluctuations are relatively small, the contribution ofthe second term is negative, leading to shorter queues.On the other hand, when CV > 1, i.e., when servicetime fluctuations are relatively large, the contributionof the second term is positive, leading to longer queues.Importantly, there can be a “piling up” of jobs due tohigh service time variability. This can happen, for ex-ample, if short service times are occasionally followed byextremely-long service times. In such situations, CV 2 islarge, resulting in long queues, even in low utilization.It is thus apparent that fluctuations in the service timeplay a central role in the behavior of M/G/1 queues.

Finally, we recall that the mean waiting time 〈T 〉 of ajob in the queue, i.e., the time elapsed from arrival to theend of service, follows from Eq. (1) by using Little’s law[14] which states 〈T 〉 = λ−1 〈N〉. Thus, we have

〈T 〉 =1/µ

1− ρ+

ρ/µ

2(1− ρ)

(CV 2 − 1

), (2)

and note that similar to the mean number of jobs, themean waiting time crucially depends on fluctuations inservice time.

To deal with the detrimental effect caused by servicetime fluctuations several different strategies have beendeveloped. When considering single server queues, themain tools at our disposal are scheduling policies (a.k.aservice policies) [102]. A scheduling policy is the ruleaccording to which jobs are served in a queue. For exam-ple, servers can serve jobs according to the order of their

arrival as e.g. happens in the supermarket. This sim-ple First-Come, First-Served policy (FCFS) seems fairwhen working with people as customers. However, whenconsidering queues of inanimate objects, fairness is notnecessarily the most important requirement and otherservice policies can also be considered.

The FCFS policy is an example of a non-sized-basedpolicy — the server serves jobs based on their order of ar-rivals rather than their size. However, in situations wherejobs sizes are known, one can utilize this knowledge to de-vise size-based policies such as Shortest-Job-First (SJF).According to this policy, once free, the server chooses towork on the job with the smallest size and hence also theshortest service time. In terms of the average waitingtime of a job in the queue, the SJF policy performs bet-ter than the FCFS policy [14]. This can be understoodintuitively. In the FCFS policy, all jobs that arrive aftera very large job will suffer from a long waiting time. Thiswill not happen in the SJF policy, where extremely largejobs which are rare will not be selected for service beforejobs of typical size. The latter are much more commonand since they are served first they spend less time inthe queue, resulting in smaller waiting times on averagecompared to Eq. (2).

In the SJF policy, the server chooses the smallest jobwaiting in the queue each time service is complete. Thisapproach can be further developed allowing the serverto inspect jobs’ sizes (including the one in service) con-tinuously in time—constantly searching for the job withthe shortest remaining service time. In such a queue,an arriving job can preempt the job in service if its ownservice time is lower. The service of the preempted jobis resumed later, starting from the point where it wasstopped. The above policy is called Shortest-Remaining-Processing-Time (SRPT) [54], and it was used e.g in webservers to reduce the waiting time of static HTTP re-quests whose sizes have been shown to follow a heavy-tailed distribution [55, 56]. It was proved in [57], thatin single-server queues where the service time of jobs isknown and preemption is possible, the SRPT policy isthe optimal policy with respect to minimizing the meanwaiting time.

Having readily available knowledge of job sizes andtheir remaining service times, as well as the ability topreempt jobs, is a luxury not shared by all queueing sys-tems. For example, in non-deterministic algorithms, e.g.,stochastic optimization methods, inputs of similar sizecan render significantly different run times [58], and thesame can happen for consecutive runs of the algorithmusing the same input. Sized-based scheduling policies likethe SJF and SRPT are then inapplicable, but we willhereby show that the intrinsic randomness of the serviceprocess can be exploited via service resetting to achievesimilar performance goals.

4

III. M/G/1 QUEUES WITH SERVICERESETTING

In Sec. II we reviewed the non-trivial dependence, ofthe mean number of jobs in an M/G/1 queue, on ser-vice time fluctuations. Namely, large relative fluctuationslead to long queues. In this section, we analyze the ef-fect of service resetting on the mean and variance of theservice time.

To understand how restarting service affects the overallservice time of a job, we consider two extreme scenarios.First, consider a situation where fluctuations in servicetimes are extrinsic to the server. Thus, imagine a serverthat serves jobs of variable size at a constant rate. In thiscase, the service time is determined exclusively by the jobsize, i.e., the bigger the job the longer the service time,and vice versa. An example of such would be a supermar-ket cashier counter. The teller serves the customers ata (roughly) constant rate, and the service time is deter-mined by the number of items each customer has. Thus,the fluctuations in service time originate solely from thevariability in the number of items each customer brought.Imagine now that the teller decides to restart service fromtime to time. Since restart does not affect the number ofitems one has, this strategy is clearly detrimental. Thetime already spent in service is lost, while the customer’srequired service time remains unaltered. Thus, restartresults in wasted time and delays.

Now, consider a queue in which fluctuations in servicetimes are intrinsic to the server, i.e., a queue in which alljobs are (roughly) identical but the time it takes to servea job is nevertheless random. Example of such would be acomputer server which runs a stochastic algorithm. Suchalgorithm employs probabilistic approaches to the solu-tion of mathematical problems, e.g., optimization. Therun time of such an algorithm can vary considerably be-tween runs. Importantly, stochastic fluctuations in runtimes come from the probabilistic solution method, andwould hence differ even between identical instances of thesame problem. Thus, resetting such an algorithm in itscourse of action would result in a new and random runtime, contrary to the case of a teller in a supermarket.

Another example of a queue in which fluctuations inservice time are intrinsic to the server can be found inenzymatic reactions. In the context of enzymes, sub-strate molecules can be viewed as costumers, forminga ‘waiting line’ to the enzyme which acts as a server.Substrate molecules are identical, and require the sametype of service: a catalytic process which converts a sub-strate molecule to a product molecule. Yet, at the singlemolecule level, chemistry is stochastic as thermal fluc-tuations render the service (catalysis) time random. Itis often the case that a substrate molecule unbinds theenzyme without completing service [81]. In this case, ser-vice is reset, and a new and random service time is drawnupon rebinding.

Motivated by the examples above, we will now considerqueues in which service time fluctuations are intrinsic to

the server. As explained above, in these queues reset-ting results in a newly drawn service time that is addedto the time already spent in service. In what follows,we will quantify the effect of resetting on the mean andvariance of the total service time. We will then go onto show that in certain situations resetting significantlyexpedites service, thus improving overall performance byshortening queues. This effect will be discussed in thenext section.

Let us consider an M/G/1 queue with service reset-ting. Assume that both the service and restart times aretwo independent and generally distributed random vari-ables. The total service time under restart, SR, is thendescribed by the following renewal equation

SR =

S if S < R ,

R+ S′R if R ≤ S ,(3)

where R is the random resetting time drawn from a dis-tribution with density fR(t), and S′R is an independentand identically distributed copy of SR. To understandthis equation, observe that when service occurs beforerestart, SR = S. However, if service is restarted at atime R ≤ S, then a new service time is drawn, and ser-vice starts over. In this case, SR = R+ S′R.

Service can be seen as a first passage process whichends when a job is served. A comprehensive frameworkfor first passage under restart was developed in [68, 69].In the following, we show how the results obtained therecan be used to gain insight on the performance of anM/G/1 queue under restart. Starting from Eq. (3), onecan obtain the probability density of SR in Laplace space(Appendix A). Here, we will only be interested in the firsttwo moments, which are given by (Appendix A)

〈SR〉 =〈min(S,R)〉Pr(S < R)

, (4)

〈S2R〉 =

〈min(S,R)2〉Pr(S < R)

+2Pr(R ≤ S)〈Rmin〉〈min(S,R)〉

Pr(S < R)2,

(5)

where min(S,R) is the minimum between S and R,Pr(S < R) is the probability of service being completedprior to restart, andRmin = {R|R = min(R,S)} standingfor the random restart time given that restart occurredprior to service. Finally, recall that the variance in theservice time is given by Var(SR) = 〈S2

R〉 − 〈SR〉2 whichwill be useful in the next section. Equations (4) & (5) as-sert that the mean and variance of the service time underrestart can be evaluated directly from the distribution ofthe resetting time and the distribution of the service timewithout resetting. Importantly, in cases where this can-not be done analytically, numerical methods can be usedto obtain 〈SR〉 and 〈S2

R〉 and as long as the distributionsof R and S are known or can be sampled from.

5

IV. M/G/1 QUEUES WITH POISSONIANSERVICE RESETTING

Poissonian resetting, i.e., resetting with a constantrate, has been extensively investigated [64–79, 101, 104–106] (see [80] for extensive review and [107] for discussionon the connection with the inspection paradox). As thename suggests, here resetting follows a Poisson processand the number of resetting events in a given time in-terval comes from the Poisson distribution. In this sec-tion we quantify the effect of Poissonian resetting on themean and variance of the total service time and the meanlength of an M/G/1 queue.

Consider an M/G/1 queue with a server that isrestarted at a rate r. In other words, we take the restarttime R to be an exponential random variable with mean1/r, and restart is thus a Poisson process with rate r. Themean and second moment of the service time can thenbe derived using Eqs. (4) and (5), giving (Appendix A 1)

〈Sr〉 =1− S(r)

rS(r), (6)

〈S2r 〉 =

2(r dS(r)dr − S(r) + 1

)r2S(r)2

, (7)

where S(r) =∫∞0

dt e−rt fS(t) is the Laplace transformof the service time, evaluated at the restart rate r.

The utilization of this queue is then given by ρr =λ〈Sr〉, and the squared coefficient of variation of the ser-vice time is CV 2

r = Var(Sr)/〈Sr〉2. We can now writethe mean queue length under restart by replacing ρ byρr, and CV 2 by CV 2

r , in Eq. (1). This yields

〈Nr〉 =ρr

1− ρr+

ρ2r2(1− ρr)

(CV 2

r − 1). (8)

Similarly, the mean waiting time 〈Tr〉 in the system canbe derived from Little’s law [13, 14], yielding an analo-gous result to Eq. (2).

To better understand the effect of resetting on themean queue length, consider the introduction of an in-finitesimal resetting rate δr. Utilizing Eq. (6), we thenfind

〈Sδr〉 ' 〈S〉 − δr〈S〉2

2

[CV 2 − 1

]+O(δr2). (9)

As expected, the first term on the right hand side ofEq. (9) is the mean of the original service time, i.e.,without resetting. The second term gives the first or-der correction, and note that its sign is governed by CV 2

of the original service time. Specifically, for CV 2 > 1,we have 〈Sδr〉 < 〈S〉 and vice versa. In other words, theintroduction of Poissonian resetting to an M/G/1 queuewill reduce the mean service time whenever the coeffi-cient of variation of the original service time is greaterthan unity.

In Fig. 2, we consider a representative example forthe effect of resetting on an M/G/1 queue with large

●

●

r *1 3 4 50.81.01.21.41.61.82.0

Resetting Rate

⟨Sr⟩-Solid

CVr-Dashed

FIG. 2. The mean (solid line) and CV (dashed line) ofthe service time with Poissonian resetting as a function ofthe resetting rate. Plots were made, using Eqs. (6) and(7), for an underlying service time taken from the inverseGaussian distribution whose density is given by fS(t) =√γ/2πt3e−γ(t−µ)

2/2µ2t, t > 0. Here, µ = 2.5 and γ = 0.5.Observe that the mean service time with resetting, 〈Sr〉, isminimized at an optimal resetting rate r∗ ' 2.092 which wasfound using Eq. (12). At this optimal resetting rate we haveCVr∗ = 1.

fluctuations in service time. Starting from a service timedistribution with CV > 1, we introduce resetting andplot the mean service time 〈Sr〉 vs. the resetting rate. Aspredicted by Eq. (9), we see that 〈Sr〉 initially decreases,obtaining a minimum at an optimal resetting rate r∗. Inaddition, note that at this optimal resetting rate we haveCVr∗ = 1. Remarkably, this observation is not specificto the service time distribution considered in Fig. 2, butis rather a universal property. Namely, it can be shownthat [69]

CVr∗ = 1, (10)

for any resetting rate, 0 < r∗ <∞, at which

d〈Sr〉dr

∣∣∣∣r∗

= 0. (11)

Substituting Eq. (6) into Eq. (11) yields

S2(r∗)− S(r∗)− r∗S′(r∗) = 0, (12)

which can be solved to obtain r∗. Combining Eq. (10)with the Pollaczek-Khinchin formula given in Eq. (8), wehave

〈Nr∗〉 =ρr∗

1− ρr∗. (13)

Note that the mean number of jobs at optimality is equiv-alent to the mean number of jobs in an M/M/1 queue[14], where job service times are exponentially distributedwith mean 〈Sr∗〉.

6

Comparing Eqs. (1) and (13), we observe that 〈N〉 >〈Nr∗〉. One can easily derive this inequality by substitut-ing CV > 1 into Eq. (1). This yields

〈N〉 =ρ

1− ρ+

ρ2

2(1− ρ)(CV 2 − 1) >

ρ

1− ρ

>ρr∗

1− ρr∗= 〈Nr∗〉, (14)

where in the last inequality we used the monotonicity ofthe function ρ

1−ρ on the interval [0, 1), and the optimality

of r∗, which implies 〈Sr∗〉 < 〈S〉 resulting in ρr∗ < ρ.We see that whenever CV > 1 for the original ser-

vice time, the mean number of jobs in the queue can bereduced by resetting service at an optimal rate. Whendoing so, one also sets the coefficient of variation of theoptimally restarted service time to unity. Thus, reset-ting not only shortens the mean service time but alsoreduces the relative stochastic fluctuations around thismean, hence providing a double advantage.

V. M/G/1 QUEUES WITH SHARP SERVICERESETTING

So far, we considered queues with resetting at a con-stant rate. In what follows, we consider a different ex-tensively investigated resetting strategy, namely sharp(a.k.a deterministic or periodic) resetting [69, 108–113].Strong motivation to study this strategy comes from thefact that in terms of mean performance, sharp resettingeither matches, or outperforms any resetting strategy ofthe type considered above Eq. (3) [69].

To see this, let us now consider an M/G/1 queue whereservice is reset at fixed time intervals of length τ . Thissimply means that the resetting time R in Eq. (3) is takenfrom the distribution fR(t) = δ(t−τ). Equations (4) and(5), can then be simplified to give (Appendix A 2)

〈Sτ 〉 =

∫ τ0dt qS(t)

1− qS(τ), (15)

〈S2τ 〉 =

[2(1− qS(τ))

∫ τ

0

dt tqS(t)

+ 2τqS(τ)

∫ τ

0

dt qS(t)]/ [1− qS(τ)]

2, (16)

where qS(τ) = 1 −∫ τ0dt fS(t) is the survival function

associated with the underlying service time.The utilization of this queue is then given by ρτ =

λ〈Sτ 〉, and the squared coefficient of variation of the ser-vice time is CV 2

τ = Var(Sτ )/〈Sτ 〉2. We can now obtainthe mean queue length under service resetting by replac-ing ρ by ρτ , and CV 2 by CV 2

τ , in Eq. (1). This yields

〈Nτ 〉 =ρτ

1− ρτ+

ρ2τ2(1− ρτ )

(CV 2

τ − 1). (17)

In Fig. 3, we consider a representative example for theeffect of sharp resetting on an M/G/1 queue with large

●

●

τ*0.1 0.3 0.7 0.90.81.01.21.41.61.82.0

Resetting time

⟨Sτ⟩-Solid

CVτ-Dashed

FIG. 3. The mean (solid line) and CV (dashed line) ofthe service time with sharp resetting as a function of theresetting time. Plots were made, using Eqs. (15) and(16), for an underlying service time taken from the InverseGaussian distribution whose density is given by fS(t) =√γ/2πt3e−γ(t−µ)

2/2µ2t, t > 0. Here, µ = 2.5 and γ = 0.5.Observe that the mean service time with resetting, 〈Sτ 〉, isminimized at an optimal resetting time τ∗ ' 0.501, whichwas found using Eq. (20). At this optimal resetting time wehave CVτ∗ ≤ 1.

fluctuations in service time. Starting from the same ser-vice time distribution used in Fig. 2 (CV > 1), we in-troduce resetting and plot the mean service time 〈Sτ 〉vs. the resetting time. Once again, we observe that 〈Sτ 〉obtains a minimum at an optimal resetting time τ∗. Inaddition, note that at this optimal resetting time we haveCVτ∗ < 1. As for the case of Poissonian resetting, thisobservation is not specific to the service time distributionconsidered in Fig. 3, but is rather a universal property.Namely, it can be shown that [69]

CVτ∗ ≤ 1, (18)

for an optimal resetting time τ∗ which brings 〈Sτ∗〉 to aglobal minimum. As before, τ∗ can be found by setting

d〈Sτ 〉dτ

∣∣∣∣τ∗

= 0 , (19)

which by substitution of Eq. (15) into Eq. (19), boilsdown to

qS(τ∗)− q2S(τ∗) + q′S(τ∗)

∫ τ∗

0

dt qS(t) = 0. (20)

Combining Eq. (18) with the Pollaczek-Khinchin for-mula given in Eq. (17), we have

〈Nτ∗〉 ≤ ρτ∗

1− ρτ∗. (21)

It is interesting to compare Eqs. (21) and (13). To thisend, we recall that the mean first passage time underoptimal sharp resetting is always smaller or equal thanthat obtained under optimal Poissonian resetting [69].Translating this result to the language of queueing, we

7

have 〈Sτ∗〉 ≤ 〈Sr∗〉, which implies ρτ∗1−ρτ∗

≤ ρr∗1−ρr∗

, by

monotonicity. We thus have

〈Nτ∗〉 ≤ 〈Nr∗〉. (22)

Equation (22) is of great importance as it reveals thatsharp resetting offers an additional improvement com-pared to the Poissonian resetting strategy. Once again,we see that whenever CV > 1 for the original servicetime, the mean number of jobs in the queue can be re-duced by resetting service at an optimal time. Whendoing so, one also reduces the coefficient of variation ofthe optimally restarted service time below unity. Thus,sharp resetting further shortens the mean service time,while also reducing the relative stochastic fluctuationsaround this mean compared to Poissonian resetting. Wenow set out to illustrate the results obtained in sectionsIV and V on several case studies.

VI. EXAMPLES

To demonstrate the power of our approach, we nowconsider two well-known service time distributions: log-normal [114–116] and Pareto [117–121], which are nowwell documented in queuing theory literature [122–131].In what follows, we first describe the effect of restart inthe case of log-normal service times and discuss Paretoservice times next.

A. Log-normal service time

Consider the case of on an M/G/1 queue whose servicetime S is log-normally distributed

fS(t) =1√

2πσte−

(ln t−µ)2

2σ2 , (23)

for t > 0, where µ ∈ (−∞,∞) and σ > 0. The survivalfunction is then given by

qS(t) = Pr(S > t) =

{12Erfc

(−µ−log(t)√

2σ

)t > 0

1 t ≤ 0(24)

The mean and variance of the service time in this caseare given by

〈S〉 = eµ+σ2

2 , (25)

Var(S) =(eσ

2

− 1)e2µ+σ

2

, (26)

such that

CV 2 = eσ2

− 1 , (27)

which is independent of µ. In the discussion below, weset the mean service time 〈S〉 to be fixed, and vary σ,which controls the relative magnitude of fluctuations inservice time via Eq. (27). Setting 〈S〉 and σ, µ is uniquelydetermined by Eq. (25).

●●

●●

0 0.5 1 1.5 20.85

0.90

0.95

1.00r* 1/τ*

Resetting rate, reciprocal resetting time

Meanservicetim

e

Sharp

Poisson

FIG. 4. The mean service time with Poissonian (orange) andsharp (blue) resetting, as a function of the resetting rate andreciprocal resetting time. Plots were made, using Eqs. (6)and (15), for an underlying service time taken from the log-normal distribution whose density is given by Eq. (23). Here,〈S〉 = 1 and σ = 1.05, yielding µ = −0.28125. The optimalresetting rate, r∗, and resetting time, τ∗, are indicated.

For Poissonian resetting, the mean service time is givenby Eq. (6), which requires the Laplace transform of thelog-normal distribution. The latter, does not have an an-alytical closed-form, but it can be evaluated numericallyfor any choice of parameters. In the case of sharp reset-ting, the mean service time is given by Eq. (15), whichrequires the survival function qS(t) given in Eq. (24).Here too, the mean service under resetting can be com-puted by numerical evaluation of the required integrals.In Fig. 4, we set 〈S〉 = 1 and σ = 1.05, and plot themean service time under Poissonian and sharp resetting.In both cases, a minimum is obtained at an optimal re-setting rate or time, depending on the resetting scheme.Observe that the optimal mean service time under sharpresetting is indeed lower than that obtained for Poisso-nian resetting.

To find the optimal resetting rate in Fig. 4, we solve

H(〈S〉, σ, r∗) = 0, (28)

where H(〈S〉, σ, r∗) denotes the left hand side of Eq. (12).This gives r∗ ' 0.885 for the given parameters. A simi-lar minimization procedure can be carried out for sharpresetting. Substitution of the survival function for thelog-normal distribution given in Eq. (24) into Eq. (20)yields the following equation for the optimal resettingtime τ∗

F(〈S〉, σ, τ∗) = 0, (29)

where F(〈S〉, σ, τ∗) denotes the left hand side of Eq. (20).This gives τ∗ ' 0.857.

Once in hand, the optimal resetting rate r∗ can be sub-stituted into Eqs. (6) and (7), to obtain the mean andvariance of the service time under optimal Poissonian re-setting. Similarly, τ∗ can be substituted into Eqs. (15)and (16), for the mean and variance of the service timeunder optimal sharp resetting. Once these quantities

8

0 0.5 1 1.5-10-2

0

ℋ(⟨

S⟩,σ

,r)

Resetting rate

(a)0 1 2 3 4 5 6

-0.2

0.0

0.2

0.4

Resetting time

ℱ(⟨

S⟩,σ

,τ)

CV≃0.87

CV≃1.03

CV≃1.21

CV≃1.42

CV≃1.66

●●

●●

●●

●●●●

●●

●●

●●

●●

●●

●●

●●

●●(b)

0 1 2 3

5

10

15

20

25

30

Squared coefficient of variation

Meannumberofjobs

Poissonian

Sharp

PKformula

FIG. 5. Panel (a): Solutions of Eqs. (28) and (29) for the Log-normal service time distribution under Poissonian (inset) andsharp resetting (main). Here, we fix 〈S〉 = 1 for the mean service time without resetting and vary σ to control relative stochasticfluctuations, CV , via Eq. (27). The intersection points of the curves with the dashed horizontal line going through the origingive the optimal restart times τ∗ (optimal restart rates r∗ in the inset). Note that higher values of σ yield lower optimalresetting times (higher resetting rates). Panel (b): The mean number of jobs in the queue as a function of the underlying CV 2

of the service time distribution. The Pollaczek-Khinchin formula gives the familiar linear dependence of Eq. (1). Also plottedare the behaviours for the optimal Poissonian and sharp resetting protocols, with colored circles matching their counterpartsin panel (a). Strong deviations from the Pollaczek-Khinchin behaviour of the non-restarted case are observed. While resettingprovides no advantage for low CV values, it can drastically reduce queue lengths when CV is high.

are known, and given the arrival rate λ, the Pollaczek-Khinchin formula can be used directly to obtain the themean number of jobs in a queue with optimal serviceresetting.

In Fig. 5, we extend the analysis presented in Fig. 4.Setting 〈S〉 = 1, we compute the optimal resetting rateand time for various values of the parameter σ (Fig. 5a),which controls the relative fluctuations in the log-normalservice time via Eq. (27). We then compare betweenthe mean number of jobs in a queue without service re-setting to that obtained when optimal Poissonian andsharp resetting are applied (Fig. 5b). We observe thatresetting can drastically shorten queues when stochas-tic fluctuations in the underlying service time are large(high CV ). To this end, note that optimal sharp resettingoutperforms optimal Poissonian resetting as predicted byEq. (22). Also, note that for high CV values the meanqueue lengths at optimal resetting drop below the meanqueue length in the absence of service time fluctuations(CV = 0). This remarkable feature illustrates how reset-ting turns the problem of service time fluctuations intoan advantageous benefit.

B. Pareto service time

We now consider the case of on an M/G/1 queue whoseservice time S is Pareto distributed

fS(t) =αLα

tα+1, (30)

for t ≥ L, where α,L > 0. The survival function is givenby

qS(t) =

{(tL

)−αt ≥ L

1 t < L. (31)

The mean and variance of the service time in this caseare given by

〈S〉 =

{∞ for α ≤ 1αLα−1 for α > 1

, (32)

Var(S) =

{∞ for α ≤ 2

αL2

(α−1)2(α−2) for α > 2, (33)

such that

CV 2 =1

α(α− 2), (34)

which is independent of L. Similar to the previous ex-ample, we now set the mean service time 〈S〉 to be fixed,and vary α, which controls the relative magnitude of fluc-tuations in service time via Eq. (34). Setting 〈S〉 and α,L is uniquely determined by Eq. (32).

For Poissonian resetting, the mean service time is givenby Eq. (6), which requires the Laplace transform of thePareto distribution. This is given by

S(r) = α(Lr)αΓ(−α,Lr), (35)

where Γ(a, x) is the upper incomplete gamma function

Γ(a, x) =

∫ ∞x

dt tα−1e−t. (36)

9

By substituting the above into Eqs. (6) and (7), we getthe following expressions for the mean

〈Sr〉 =1− α(Lr)αΓ(−α,Lr)αr(Lr)αΓ(−α,Lr)

, (37)

and second moment

〈S2r 〉 =

2α(α− 1)(Lr)αΓ(−α,Lr)− 2αe−Lr + 2

(αr)2(Lr)2αΓ(−α,Lr)2,(38)

of the service time under Poissonian resetting.In the case of sharp resetting, the first two moments of

the service time are given by Eqs. (15) and (16). Substi-tuting the survival function (31) into Eqs. (15) and (16)yields

〈Sτ 〉 =Lα

α− 1+

Lα− τ(α− 1)

[(τL

)α − 1] , (39)

〈S2τ 〉 =

2τ2Lα (Lα − (α− 1)τα)

(α− 2)(α− 1) (τα − Lα)2 +

2ατα+1Lα+1

(α− 1) (τα − Lα)2

+αL2τα

(α− 2) (τα − Lα). (40)

Note that in the above we only considered resetting timesτ > L as the support of the Pareto distribution is givenby t ≥ L. Also, note that Eq. (39) is valid for α 6= 1.Similarly, in Eq. (40) we require α 6= 1, 2. These sin-gular cases require separate treatment, following similarfootsteps.

In Fig. 6, we set 〈S〉 = 1 and α = 2.1, and plot themean service time under Poissonian and sharp resetting.In both cases, a minimum is obtained at an optimal re-setting rate or time, depending on the resetting scheme.Once again, the optimal mean service time under sharpresetting is indeed lower than that obtained for Poisso-nian resetting.

To find the optimal resetting rate in Fig. 6, we solve

S(〈S〉, α, r∗) = 0, (41)

where S(〈S〉, α, r∗) denotes the left hand side of Eq. (12),after substituting the Laplace transform of Eq. (35). Thisgives r∗ ' 0.034. A similar minimization procedure canbe carried out for sharp resetting. Substitution of the sur-vival function for the Pareto distribution given in Eq. (31)into Eq. (20) yields the following equation for the optimalresetting time τ∗

G(〈S〉, α, τ∗) = τ∗, (42)

where

G(〈S〉, α, τ) =( τL

)α (−ατ + α2L+ τ

). (43)

Solving gives τ∗ ' 1.990.The optimal resetting rate r∗ can be substituted into

Eqs. (6) and (7), to obtain the mean and variance of theservice time under optimal Poissonian resetting. Simi-larly, τ∗ can be substituted into Eqs. (15) and (16), forthe mean and variance of the service time under optimal

●●

●●

0.01 0.1 0.5 1

0.95

1

1.05

r* 1/τ*

Resetting rate, reciprocal resetting time

Meanservicetim

e

Sharp

Poisson

FIG. 6. The mean service time with Poissonian (orange) andsharp (blue) resetting, as a function of the resetting rate andreciprocal resetting time. Plots were made, using Eqs. (37)and (39), for an underlying service time taken from the Paretodistribution whose density is given by Eq. (30). Here, 〈S〉 = 1and α = 2.1, yielding L ' 0.524. The optimal resetting rate,r∗, and resetting time, τ∗, are indicated.

sharp resetting. Once these quantities are known alongwith the arrival rate λ, the Pollaczek-Khinchin formulacan be used directly to obtain the the mean number ofjobs in a queue with optimal service resetting.

In Fig. 7, we extend the analysis presented in Fig.6. Setting 〈S〉 = 1, we compute the optimal resettingrate and time for various values of the parameter α (Fig.7a), which controls the relative fluctuations in the Paretoservice time via Eq. (34). We then compare betweenthe mean number of jobs in a queue without service re-setting to that obtained when optimal Poissonian andsharp resetting are applied (Fig. 7b). Like in the pre-vious example, here too, we observe that resetting caninduce shorter queues when the underlying service timevariability (CV ) is large. Noteworthy in this case is thedominance of sharp resetting over Poissonian resetting aspredicted by Eq. (22). Moreover, for high CV values themean queue length at optimal resetting drops below themean queue length in the absence of service time fluc-tuations (CV = 0). This hallmark property shows howresetting can take advantage of large stochastic fluctua-tions in service time and expedite the process completion.

VII. CONCLUSIONS AND OUTLOOK

Regulating the number of jobs in a queue is an integralpart of performance modeling and optimization of queu-ing systems. One problem that arises in this context isthat large stochastic fluctuations in service times lead tosignificant backlogs and delays. In this paper, we showedhow this problem can be mitigated by service resetting.Indeed, we have shown that when applied to servers withintrinsically highly service time variability, resetting candramatically reduce queue lengths and job waiting times.

Our analysis was based on the canonical M/G/1 queu-ing model in which jobs arrive to the queue following

10

0.005 0.02-2.×10-5

02.×10-5

(⟨

S⟩,α

,r)

Resetting rate

(a)

2.0 2.1 2.2 2.3 2.4-2

02468

10

Resetting time

(⟨

S⟩,α

,τ) CV≃1.20

CV≃1.33

CV≃1.51

CV≃1.76

CV≃2.18

●●●●●●●●●●

●●●●●●●●●●

(b)

1 2 3 40

5

10

15

20

25

30

Squared coefficient of variation

Meannumberofjobs

Poissonian

Sharp

PKformula

FIG. 7. Panel (a): Solutions of Eqs. (41) and (42) for the Pareto service time distribution under Poissonian (inset) and sharpresetting (main). The intersection points of the curves for different α (i.e., for different CV ) with the dashed horizontal linethrough the origin give the optimal restart time τ∗ (optimal restart rate r∗ in the inset). Panel (b): The mean number of jobsin the queue as a function of the underlying CV 2 of the service time distribution. The Pollaczek-Khinchin formula gives thefamiliar linear dependence of Eq. (1). Also plotted are the behaviours for the optimal Poissonian and sharp resetting protocols,with colored circles matching their counterparts in panel (a). Strong deviations from the Pollaczek-Khinchin behaviour of thenon-restarted case are observed. While resetting provides no advantage for low CV values, it can drastically reduce queuelengths when CV is high.

a Poisson process and service times come from a gen-eral distribution. The renowned Pollaczek-Khinchin for-mula (Eq. 1) asserts that the mean number of jobs inthis queue grows linearly with the squared coefficient ofvariation, CV 2, of the service time. Employing the re-cently developed framework of first-passage under restart[68, 69] to the M/G/1 queue, we showed that Poissonianservice resetting reduces both the mean and variance ofthe overall service time when CV > 1, i.e., exactly whenservice time fluctuations start to become a major sourceof concern. Sharp service resetting, a.k.a deterministic orperiodic, performs even better and could in some caseslower the mean and variance of the service time evenwhen CV < 1. In both cases, the net result is muchshorter queues and examples were given to show thatmean queue lengths can drop well below those attainedfor servers with no service time fluctuations (determin-istic service time, CV = 0). Service resetting can thusturn a well-known drawback of queueing systems into afavourable advantage.

Our work is the first step towards the application of re-setting as a fluctuations mitigation strategy in queueingsystems. Some future research directions are discussedbelow. The analysis presented above yielded an analyti-cal result for the mean number of jobs in an M/G/1 queueunder service resetting, thus generalizing the Pollaczek-Khinchin formula to this case. In the future, it wouldbe interesting to generalize this result beyond the mean,so as to give the full distribution of the queue length un-der service resetting. Another interesting direction wouldbe to generalize the analysis to other queueing systems,e.g., the M/G/k queue which has k-servers or to othermulti-server systems.

Going to queues with non-Markovian arrivals is alsoof interest, as realistic job arrival processes need not be

Poissonian. To this end, we recall Kingman’s approxima-tion formula [132], which asserts that the G/G/1 queuesuffers from the same linear dependence of the meannumber of jobs on the squared coefficient of variationof the service time. Thus, applying service resetting tothis queue will qualitatively have the same effect as in thesimpler M/G/1 queue. Moreover, as the arrival process isdecoupled from the service process, the results derived inSecs. III-V above can be applied directly to the G/G/1queue, with the only difference being Kingman’s approx-imation formula substituting for Pollaczek-Khinchin. In-deed, this can be done as both formulas only depend onthe first two moments of the service time distribution.Conclusions coming from this work thus apply broadlyto single-server queuing systems, and their ramificationson multi-server queues and queue networks remain to beworked out and understood in detail.

ACKNOWLEDGMENTS

Arnab Pal is indebted to Tel Aviv University (via theRaymond and Beverly Sackler postdoc fellowship and thefellowship from Center for the Physics and Chemistryof Living Systems) where the project started. ShlomiReuveni acknowledges support from the Israel ScienceFoundation (grant No. 394/19). This project has re-ceived funding from the European Research Council(ERC) under the European Union’s Horizon 2020 re-search and innovation programme (Grant agreement No.947731).

11

Appendix A: Service time under restart

In this section, we briefly review a few essential resultsthat were derived employing the framework of first pas-sage under restart in [69]. We start by recalling thatthe distribution of service time under restart in Laplacespace reads [69]

SR(s) =Pr(S < R)Smin(s)

1− Pr(R ≤ S)Rmin(s), (A1)

where SR(s) =∫∞0

dt e−st fSR(t). Moreover, we havedefined two auxiliary random variables

Rmin ≡ {R|R ≤ S} ,Smin ≡ {S|S < R} . (A2)

In words, Rmin is the restart time conditioned on theevent that restart occurs before the service is over. Sim-ilarly, Smin is the time conditioned that service occurredprior to a restart. The probability density functions ofRmin and Smin are given by [69]

fRmin(t) =fR(t)

∫∞t

dt′ fS(t′)

Pr(R ≤ S)=fR(t)Pr(S > t)

Pr(R ≤ S),

(A3)

fSmin(t) =

fS(t)∫∞t

dt′ fR(t′)

Pr(S < R)=fS(t)Pr(R > t)

Pr(S < R).

(A4)

The moments can now easily be computed from Eq. (A1)by noting that

〈SnR〉 = (−1)ndn

dsnSR(s)|s→0, (A5)

which gives [69]

〈SR〉 =〈min(S,R)〉Pr(S < R)

(A6)

〈S2R〉 =

〈min(S,R)2〉Pr(S < R)

+2Pr(R ≤ S)〈Rmin〉〈min(S,R)〉

Pr(S < R)2,

(A7)

where 〈Rmin〉 can be computed directly from Eq. (A4).The other components can also be systematically derivedwith the knowledge of individual time densities. For ex-ample, the numerator 〈min(S,R)〉 in Eq. (A6) is givenby

Pr(min(S,R) ≤ t) = 1− Pr(S > t)Pr(R > t). (A8)

and the denominator in Eq. (A6) is given by

Pr(S < R) =

∫ ∞0

dt fR(t)Pr(S < t)

=

∫ ∞0

dt fR(t)

∫ t

0

dt′ fS(t′). (A9)

1. Moments of service time under Poissonianresetting

Consider the case of Poissonian resetting. When therestart time R is Poissonian, its probability density func-tion is given by

fR(t) = r e−rt, (A10)

and the cumulative distribution function is given by

Pr(R ≤ t) = 1− e−rt, (A11)

where r is the resetting rate.In the case of Poissonian resetting, the cumulative dis-

tribution function of min(S,R) can be written as followed

Pr(min(S,R) ≤ t) = 1− Pr(S > t) e−rt. (A12)

Using the following formula for the expectation value ofnon-negative random variable

〈X〉 =

∫ ∞0

dt qX(t), (A13)

where qX(t) = Pr(X > t) is the survival function of X,one can easily show that

〈min(S,R)〉 =

∫ ∞0

dt Pr(S > t)Pr(R > t)

=

∫ ∞0

dt Pr(S > t)e−rt

=

∫ ∞0

dt [1− Pr(S < t)] e−rt

=

∫ ∞0

dt e−rt −∫ ∞0

dt Pr(S < t)e−rt

=1

r−∫ ∞0

dt e−rt(∫ t

0

dt′ fS(t′)

)=

1

r− S(r)

r=

1− S(r)

r, (A14)

where in the second to last step we used a known for-mula for the Laplace transform of a time-domain integra-

tion:∫∞0dt(∫ t

0dτ g(τ)

)e−rt = g(r)

r , where g(r) is the

Laplace transform of g(t). Similarly, we can use Eq. (A9)to find

Pr(S < R) =

∫ ∞0

dt r e−rtPr(S < t)

= r

∫ ∞0

dt e−rt Pr(S < t)

= S(r), (A15)

Substituting Eqs.(A14) and (A15) into Eq.(A6), we getthe following formula for the mean service time underPoissonian resetting

〈Sr〉 =1− S(r)

rS(r), (A16)

12

which was used in the main text (see Eq. (6)). Thisequation was previously derived in [69].

We now turn to the derivation of formula for the secondmoment of the service time under Poissonian resetting,〈S2r 〉. To do so, we start by deriving 〈min(S,R)2〉. By

taking the derivative of Eq. (A12), one can easily obtainthe probability density function of min(S,R)

fmin(S,R)(t) = fS(t) e−rt + r Pr(S > t) e−rt.(A17)

Now, we can use the probability density function givenin Eq.(A17) to calculate 〈min(S,R)2〉

〈min(S,R)2〉 =

∫ ∞0

dt t2(fS(t) e−rt + r Pr(S > t) e−rt

)=

∫ ∞0

dt t2fS(t) e−rt

+ r

∫ ∞0

dt t2 Pr(S > t) e−rt

=d2S(r)

dr2+ r

∫ ∞0

dt t2 (1− Pr(S < t)) e−rt

=d2S(r)

dr2+

2

r2− r

∫ ∞0

dt t2Pr(S < t)e−rt

=d2S(r)

dr2+

2

r2− r

d2(S(r)r

)dr2

=2r dS(r)

dr − 2S(r) + 2

r2, (A18)

where during the derivation we used a known property

of Laplace transforms:∫∞0dt tng(t)e−rt = (−1)n d

ng(r)drn ,

where n is a positive integer with g(r) being the Laplacetransform of g(t).

We now turn to calculate 〈Rmin〉. This can be explic-itly done using Eq.(A4)

〈Rmin〉 =

∫ ∞0

dt tfRmin(t)

=1

Pr(R ≤ S)

∫ ∞0

dt t re−rtPr(S > t)

=r

1− Pr(S < R)

∫ ∞0

dt t e−rt (1− Pr(S < t))

=r(∫∞

0dt t e−rt −

∫∞0dt t e−rtPr(S < t)

)1− S(r)

=

r

(1r2 +

d(S(r)r

)dr

)1− S(r)

=r dS(r)

dr − S(r) + 1

r(

1− S(r)) . (A19)

Having all the terms on the right hand side of Eq. (A7)

in hand, we can now calculate 〈S2r 〉

〈S2r 〉 =

〈min(S,R)2〉Pr(S < R)

+2Pr(R ≤ S)〈Rmin〉〈min(S,R)〉

Pr(S < R)2

=2r dS(r)

dr − 2S(r) + 2

r2S(r)

+2(

1− S(r))rdS(r)dr −S(r)+1

r(1−S(r))1−S(r)

r

S(r)2

=2(r dS(r)dr − S(r) + 1

)r2S(r)2

, (A20)

which was used in the main text (see Eq. (7)).

2. Moments of service time under sharp resetting

Consider now the case of deterministic resetting.When the restart time R is deterministic, its probabil-ity density function is given by

fR(t) = δ(t− τ), (A21)

where τ is the resetting time and δ(t) is the delta func-tion. In the case of sharp resetting, the cumulative distri-bution function of min(S,R) can be written as followed

Pr(min(S, τ) ≤ t) = 1− Pr(S > t)θ(τ − t), (A22)

where we have used Pr(R > t) =∫∞tdt′ δ(t′ − τ) =

θ(τ − t), where θ is the Heaviside step function. UsingEq. (A13) once again one can easily show that

〈min(S, τ)〉 =

∫ ∞0

dt Pr(S > t)θ(τ − t) =

∫ τ

0

dt qS(t),

(A23)

where qS(τ) = 1 −∫ τ0dt fS(t) is the survival function

associated with the underlying service time. To derivePr(S < R) we once again use Eq. (A9) to find

Pr(S < R) =

∫ ∞0

dt δ(t− τ)Pr(S < t)

=

∫ ∞0

dt δ(t− τ)(1− qS(t))

= 1− qS(τ). (A24)

Substituting Eqs.(A23) and (A24) into Eq.(A6), we getthe following formula for the mean service time undersharp restart

〈Sτ 〉 =

∫ τ0dt qS(t)

1− qS(τ), (A25)

which was used in the main text (see Eq. (15)). Thisequation was previously derived in [108, 111]. A discreteanalog for Eq. (A25) was derived in [113].

13

We now turn to the derivation of formula for the secondmoment of the service time under sharp restart, 〈S2

τ 〉.To do so, we start by deriving 〈min(S, τ)2〉. By takingthe derivative of Eq. (A22), one can easily obtain theprobability density function of min(S, τ)

fmin(S,τ)(t) = −∂qS(t)

∂tθ(τ − t) + qS(t)δ(τ − t). (A26)

Now, we can use Eq.(A26) to compute 〈min(S, τ)2〉 whichreads

〈min(S, τ)2〉 =

∫ ∞0

dt t2[−∂qS(t)

∂tθ(τ − t) + qS(t)δ(τ − t)

]= τ2qS(τ)−

∫ τ

0

dt t2∂qS(t)

∂t

= τ2qS(τ)− τ2qS(τ) + 2

∫ τ

0

dt t qS(t)

= 2

∫ τ

0

dt t qS(t). (A27)

We now turn to calculate 〈Rmin〉. Since R is determinis-tic, 〈Rmin〉 is simply τ . This can also be explicitly shown

using Eq.(A4)

〈Rmin〉 =

∫ ∞0

dt tfRmin(t)

=1

Pr(τ ≤ S)

∫ ∞0

dt tδ(t− τ)Pr(S > t)

=τqS(τ)

qS(τ)= τ. (A28)

Substituting Eqs. (A23), (A24), (A27) and (A28) intoEq. (A7) we arrive at the following expression

〈S2τ 〉 =

〈min(S, τ)2〉Pr(S < τ)

+2Pr(τ ≤ S)〈Rmin〉〈min(S, τ)〉

Pr(S < τ)2

=2∫ τ0dt tqS(t)

1− qS(τ)+

2τqS(τ)∫ τ0dt qS(t)

(1− qS(τ))2

=2(1− qS(τ))

∫ τ0dt tqS(t) + 2τqS(τ)

∫ τ0dt qS(t)

(1− qS(τ))2,

(A29)

which was used in the main text (see Eq. (16)).

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