Raymond A Marie
email: raymond.marie@irisa.fr

Performance of a Single Server Queue Supported by an Intermittent Server

Keywords:

This chapter concerns the situation of a queue with one regular single server supported by an additional intermittent server who, in order to decrease the mean response time, i) leaves the back office to join the first server when the number of customers reaches the threshold K, ii) leaves the front office when he has no more customers to serve. This study produces a closed form solution for the steady state probability distribution and for different metrics such as expected response times for customers or expectation of busy periods. Then, for a given value of K, the influence of the intermittent server on the response time is exhibited. The consequences on the primary task of the intermittent server are investigated through metrics such as mean working and pseudo-idle periods. Finally, a cost function is proposed from which an optimal value of the threshold K is obtained.

Introduction

Let us consider a single server queue where the server can be supported by a second one who i) leaves his current work to join the first server when the number of customers reaches a threshold K, ii) leaves the queuing system when he has no more customers to serve. A typical example of such a situation comes from the banking sector where the unique server from the front office is supported by a second server regularly assigned to the back office who joins the front office as soon as the number of customers reaches a given threshold (denoted here by the integer K). But such a situation could come from a more industrial area. The introduction of an intermittent server allows to decrease the expected waiting times of customers at a lower cost than affecting an extra permanent server. And the aim of this study is to determine the efficiency of such a policy.

Note that a closed situation is the one of the supermarket check-out counters where a counter can be activated/deactivated based on the states of the different queues. This larger model is a good example to be used in a course on discrete event simulation as a practical exercise because the queuing model is easy to elaborate and has no (known) analytical solution in its general configuration. This help students to realize all the advantages of a simulation approach. In addition, such a model is easily adaptable to other fields such as those of telecommunication or of data centers. Nevertheless, when possible, an analytical solution must be looked for since its cost is generally lower than the one of the simulation approach.

Although most of the research work in the domain of the M/M/r queue with intermittent servers has been done through the use of simulation, we noted some developments connected to the subject. In 1971, J. Blackburn published a report [START_REF] Joseph D Blackburn | Optimal control of queueing systems with intermittent service[END_REF] relative to a M/G/1 queue in which the server is an intermittent one who starts working when the number of customers crosses some threshold. This threshold is the value realizing the optimum of an objective function. A more recent analytical study investigated the case of an airline check-in counters set in an airport [START_REF] Parlar | Dynamic allocation of airline check-in counters: a queueing optimization approach[END_REF]. In this study, Parlar et. al elaborated a Markovian model and its transient solution. A major difference with the supermarket check-out system is that the number of customers to be served is known in advance (number of customers who have a reserved seat for a given flight). The problem is to control the number of open check-in counters such that all the customers that will show up before a deadline T will be served on time (such that the plane can take off on time). But most of the literature involving intermittent servers concerns studies where the activations of the servers depend on reliability/availability of the set of servers rather than on the states of the systems.

Another related class of models is the "coupled processor model" where each processor can help the other when it is idle. The two queues have their own arrival processes and service time distributions. Such a class has been the object of intensive analytical works in the past. Close to that is the case where the behaviors of the servers are no more symmetrical and only one processor can, when it becomes idle, give time to the other processor until its own queue reaches a given threshold (see the intensive study of Osogami et al. [START_REF] Osogami | Analysis of cycle stealing with switching times and thresholds[END_REF]). Note also the different model known as "the slow server problem" (see [START_REF] Rubinovitch | The slow server problem: a queue with stalling[END_REF]) where, depending on the values of the parameters, the use of the slow server may increase the response time.

The present study is different in the sense that the server who gives some part of his time is not idle but works on tasks which are not directly impacting customers (the notion of response time is in some sense meaningless). This study is less general than the one cited above ( [START_REF] Osogami | Analysis of cycle stealing with switching times and thresholds[END_REF]) but produces a closed form solution for the steady state probability distribution and for different metrics such as expected waiting times for customers or expectation of busy periods for the intermittent server. Our objective is to promote a better understanding of the benefits of such a strategy. In particular, we have to consider the trade-off between the help to the customer and the perturbation of the work in the back office. This is achieved thanks to a cost function providing an optimal value of the threshold K as a tool to help a manager in charge of the economical decision.

The paper is organized as follows: in Sect. 2 we present a Markovian model of the investigated system while in the following section we exhibit the steady state probability distribution of the stochastic process and the expression of the mean number of customers (or mean response time) in terms of the different parameters. In Sect. 4, we conduct the determination of the expectation of the time spent by the second server in one passage in the back office and those of the expectation of one sojourn time at the front office. In the following section we introduce a cost function allowing us to provide an optimal threshold K * . Finally, we conclude by summarizing the advantages of using an intermittent server (Sect. 6).

Hypotheses and Model

We consider that the two servers are equivalent and that the service times are independent and identically distributed random variables following an exponential distribution with rate µ. The first server assigned to the front office stays available for serving the arriving customers.

When there are (K -1) customers, if the server affected to the back office is not already serving in the front office, then this server leaves the back office at the instant of arrival of a new customer and starts serving him in the front office. Once he is in the front office, the second server stays there until he has no more customers to serve and re-integrates the back office.

We assume the customer arrival process is Poisson with rate λ.

Under these hypotheses, the stochastic process modeling the number of customers in the office is a continuous time Markov chain (CTMC) {X(t), t ≥ 0} ([2], [START_REF] Harrison | Performance Modelling of Communication Networks and Computer Architecture[END_REF], [START_REF] Sahner | Performance and Reliability Analysis of Computer Systems: An Example-Based Approach Using the SHARPE Software Package[END_REF]). Its transition graph is given in Figure 1.

A couple (i, 0) (respectively (i, 1)) denotes a state where i customers are present and where the second server is in the back office (respectively present). State (0) refers to the empty system and, for i ≥ K, state i denotes the system when i customers and the second server are present. Note that the first server is idle in state (1, 1). In addition, E 0 (respectively E 1 ) will denote the subset of states where the second server is in the back office (respectively present):

0 1,1 2,1 i K K+1 K-1,1 K-1,0 1,0 2,0 λ λ λ λ λ λ λ λ λ λ λ λ λ µ µ µ µ µ µ µ 2µ 2µ 2µ 2µ 2µ 2µ 2µ • • • • • • • • • • • •
E 0 = {(0), (1, 0), ... (K -1, 0)}, E 1 = {(1, 1), ...(K -1, 1), (K), (K + 1), ...} .
The steady state probability distribution of this CTMC is determined in the following section.

Note that the case K = 2 corresponds to a M/M/2 queue with a little specificity: once the queue is empty, the first server deals with the new arrival, the second server arriving only when a new arrival finds the first server busy, and going back as soon as there is no more customer to serve in the front office. But from the customer point of view, this specificity does not affect the performance of the queue.

Steady State Probability Distribution, Mean Number of Customers

Steady State Probability Distribution

For any state e, π e will denote the steady state probability of state e. Defining ρ = λ/2µ, we note that the steady state probability will exist only if ρ < 1. Using the Chapmann-Kolmogorov (C-K) equations of states (i, 0), i = 2, ..., K -1, it is not difficult to prove by induction the relation:

π K-i,0 =   i-1 j=0 φ j   π K-1,0 , i = 2, ..., K -1 , (1)
where φ = µ/λ. Use of the cut theorem on the partition {E 0 , E 1 } and of the steady state C-K equation of state (1, 1) gives us

π 1,1 = 1 (1 + 2φ) π K-1,0 . (2)
Then, using equations ( 2) and ( 1) and the C-K equation for state (0), we can express probability π K-1,0 in term of π 0 as :

π K-1,0 = π 0   φ (1 + 2φ) + φ   K-2 j=0 φ j     -1 , (3)
or, for the case φ = 1, as :

π K-1,0 = π 0 (1 + 2φ)(1 -φ) D 0 , (4)
where

D 0 = φ[(1 -φ) + (1 + 2φ)(1 -φ K-1 )]
. Considering now the C-K equations of states (i, 1), i = 2, ..., K -1, we can prove by induction that :

π i,1 = π 1,1 (1 + ρ) -2ρ i (1 -ρ) i = 2, ..., K , (5)
Since ρ = 1 is a root of the numerator, let us note that this probability can also expressed as :

π i,1 = π 1,1   1 + 2 i-1 j=1 ρ j   i = 2, ..., K , (6)
When i = K, we get in particular the probability π K,1 that we can rename π K without any ambiguity :

π K = π 1,1 (1 + ρ) -2ρ K (1 -ρ) = π 1,1   1 + 2 K-1 j=1 ρ j   . (7)
Then, using equations ( 2) and ( 4), we express the probability π K as a function of probability π 0 (again for the case φ = 1) :

π K = π 0 (1 + ρ) -2ρ K (1 -ρ) (1 -φ) D 0 . ( 8
)
Considering the probabilities π i , i > K, their expressions are easily obtained thanks to the use of the cut theorem :

π i = ρ i-K π K , i > K . (9)
Let us now consider the normalizing equation that we can write as :

S 0 + S 1 = 1 , (10)
where

S 0 = π 0 + K-1 i=1 π i,0 and S 1 = K-1 i=1 π i,1 + ∞ i=K π i .
Note that S 0 is the steady state probability that the intermittent server is working in the back office and that S 1 is the steady state probability that the intermittent server is working in the front office. This last sum S 1 will be also used latter when looking for the optimal threshold.

Using equations ( 1), ( 2), ( 4), ( 5), ( 8) and (9), we show in Sect. 7 that the probability π 0 can be written as :

π 0 = (1 -ρ)(1 -φ)D 0 D 1 , (11)
where

D 1 = (1 -ρ){φ(1 -φ) 2 + (1 + 2φ)[(K -1)(1 -φ) -φ 2 (1 -φ K-1 )]} + +(1 -φ) 2 [K + ρ(K -1)] . (12)
For the special case where φ = 1, equations ( 1), ( 2), ( 4) and ( 5) reduce to:

π K-1,0 = 3 3K -2 π 0 , and π K-i,0 = iπ K-1,0 , i = 2, ..., K -1 , (13)
π 1,1 = 1 3 π K-1,0 , and
π i,1 = (3 -2 -(i-2) )π 1,1 , i = 2, ..., K , (14)
while it is shown in Sect. 7 that probability π 0 satisfies :

π 0 = 2(3K -2) 3(K(K + 3) -2) . ( 15
)
For the case where K = 2, the transition graph of the CTMC is given on Figure 2. Some of the equations given for the general case become simpler (in particular because the expression D 0 equals 2φ(1 -φ 2 ) when K = 2) and it is not difficult to find again the well known result of the M/M/2 queue :

π 0 = (1 -ρ) (1 + ρ) . (16)
Let us remark that for ρ = 1/2, we obtain π 0 = 1/3. In that case φ = 1, and this result agrees with the one obtained thanks to relation (15) when K = 2.

3 2 4 i 0 1,0 1,1 λ λ λ λ λ λ λ λ µ µ µ µ 2µ 2µ 2µ 2µ 2µ • • • • • •

Mean Number of Customers, Mean Waiting Time

The determination of the mean number of customers IE[N ] is purely technique. For φ = 1, it is shown in Sect. 8 that this expectation satisfies the following relation :

IE[N ] = (1 -φ) D 1 (1 -ρ)(1 + 2φ) K(K + 1) 2 - K (1 -φ) + φ(1 -φ K ) (1 -φ) 2 . +(1 -φ) (1 + ρ) K(K -1) 2 + K + ρ(K -1) (1 -ρ)
When K = 2, it is not difficult to find again the well known result of the M/M/2 queue :

IE[N ] = 2ρ/(1 -ρ 2 ) . (17)
For the special case where φ = 1, it is also shown in Sect. 8 that

IE[N ] = K(K(K + 3) + 8) -4 3(K(K + 3) -2) . ( 18
)
Note that for K = 2, IE[N ] = 4/3. This result agrees with the one obtained thanks to relation (17) when ρ = 1/2, i.e., (φ = 1).

Because the aim of using an intermittent server is to decrease the waiting time of the customer in the front office, it is also interesting to consider the expected waiting time IE [W ]. For that we first obtain the expected response time by use of the Little's formula and then subtract the mean service time :

IE[W ] = 1 λ IE[N ] - 1 µ . ( 19
)
We may prefer to consider what we will call a "normalized" expected waiting time IE[W N ] by taking the mean service time (i.e., 1/µ) as the time unit. This gives us :

IE[W N ] = µIE[W ] = µ λ IE[N ] -1 = φIE[N ] -1 .
Note that the "normalized" expected waiting time has no dimension and is therefore independent of the initial time unit.

For a given value of ρ we expect that the expected number of customers is greater than the value given by the M/M/2 queue. While, as long as ρ is lower than 1/2, the expected number of customers is lower than the ratio 2ρ 1 -2ρ

, which corresponds to the value given by the M/M/1 queue with 2ρ as the utilization factor. In Figure 3, we have plotted the expectation of the number of customers as a function of ρ, for different values of the integer K. As we would expect, this expectation is increasing with ρ and with K. Note that without the second server, the mean number of customers would tend to infinity when ρ tends to 1/2.

Pseudo-idle and Busy Periods of the Intermittent Server

The pseudo-idle period of the second server is defined as the period of time during which this server is working in the back-office. We are interested by the expectation of such a period because we understand that a too short period would have a negative effect on the productivity of the server. Such a period corresponds to a sojourn time of the CTMC in the subset E 0 and therefore we need to obtain the expectation of this sojourn time.

Mean Time of a Passage in the Back Office

First let us determine the probability that a pseudo-idle period starts in state (0) (respectively in state (1, 0)). . Considering these three eventualities, we see that when the CTMC enters subset E 0 , it enters it through state (0) with probability 0.5(µ/(λ + µ)) 0.5(1 + µ/(λ + µ))

or enters it through state (1, 0) with probability 0.5 0.5(1 + µ/(λ + µ))

. These two expressions reducing respectively to φ 1 + 2φ and 1 + φ 1 + 2φ .

Let assume that X(0) = 0. Let T A be the sojourn time in the subset E 0 :

T A = inf {t|X(t) = K} .
In order to express the expectation of T A , we first consider the random variable T i defined as the time it takes to the CTMC to reach state (i + 1, 0) given X(0) = (i, 0). We also denote the expectation of T i by α i . Introducing the discrete random variable I i such that, for i ≥ 0 :

I i =       
1 if the first transition of the CTMC from state (i, 0) is a jump to state (i + 1, 0); 0 if the first transition of the CTMC from state (i, 0) is a jump to state (i -1, 0); we get when conditioning w.r.t.

I i : IE[T i |I i = 1] = 1 λ+µ , and IE[T i |I i = 0] = 1 λ+µ + α i-1 + α i .
For i = 0, we have immediately

IE[T 0 ] = 1 λ
. Since the departure rate from state (i, 0) equals (λ + µ)

while the transition rate from state (i, 0) to state (i + 1, 0) equals λ, the probability that the first transition of the CTMC from state (i, 0) is a jump to state (i+1, 0) is P(

I i = 1) = λ λ + µ
. Therefore, deconditioning the expectation α i = IE[T i ] gives us, for i > 0,

α i = 1 λ + µ λ λ + µ + 1 λ + µ + α i-1 + α i µ λ + µ , that reduces to α i = 1 λ (1 + µ α i-1 ) . Since α 0 = IE[T 0 ] = 1 λ
, we can compute successfully α 0 , α 1 ,α 2 , .... It is not difficult to prove that

α i = 1 λ i j=0 φ j .
In addition, IE[T A ] depends on the way the CMTC enters the subset

E 0 since IE[T A |X(0) = 0] = K-1 j=0 α j , while IE[T A |X(0) = (1, 0)] = K-1 j=1 α j .
Therefore, after deconditioning we obtain :

IE[T A ] = 1 λ 1 2(1 + ρ) + 1 λ (K -1) + K-1 i=1 (K -i)φ i . ( 20
)
We can scale this result by expressing this time expectation in term of a number of mean service times : In Figure 4, we have plotted the scaled expectation of the pseudo-idle period of the second server as a function of ρ, for different values of the integer K. We can say that the expectation of the pseudo-idle period of the second server is important when ρ is between 0 and around 0.4, Remember that when ρ = 0.4, the utilization factor of the single server of the M/M/1 queue equals 0.8. As we would expect, this expectation is decreasing with ρ and increasing with K.

µIE[T A ] = φ 2(1 + ρ) + φ (K -1) + K-1 i=1 (K -i)φ i . (21)
Note also that if the manager decides to change the rule by switching from K to (K + 1), then the scaled expectation will be increased of the quantity:

∆ K (µIE[T A ]) = µIE[T A (K + 1)] -µIE[T A (K)] = φ K i=0 φ i .
Even in the case where φ = 1 (i.e., ρ = 0.5), this increase can be shown to correspond to (K + 1) mean service times!

Mean Time of a Passage in the Front Office

Now let IE[T P ] be the expectation of a period spent in the front office by the intermittent server. This server starts such a period with the frequency λπ K-1,0 . Using the fact that this frequency must be equal to (IE[T A ] + IE[T P ]) -1 , we obtain a first expression for λIE[T P ] :

λIE[T P ] = [π K-1,0 ] -1 -λIE[T A ] .
Then, starting from equations ( 4) and (11) we express the inverse of probability π K-1,0 as :

[π K-1,0 ] -1 = D 1 (1 -ρ)(1 + 2φ)(1 -φ) 2 , = φ (1 + 2φ) + (K -1)(1 -φ) -φ 2 (1 -φ K-1 ) (1 -φ) 2 + [1 + (K -1)(1 + ρ)] (1 -ρ)(1 + 2φ) .
Using equation ( 20) we develop the expression of λIE[T A ] as :

λIE[T A ] = 1 2(1 + ρ) + (K -1) + K-1 i=1 (K -i)φ i = φ (1 + 2φ) + (K -1) + K K-1 i=1 φ i - K-1 i=1 iφ i = φ (1 + 2φ) + (K -1) + K 1 -φ K (1 -φ) -1 - (K -1)φ K+1 -Kφ K + φ (1 -φ) 2 = φ (1 + 2φ) + (K -1) + Kφ(1 -φ K-1 ) (1 -φ) - (K -1)φ K+1 -Kφ K + φ (1 -φ) 2 = φ (1 + 2φ) + (K -1)(1 -φ) 2 + Kφ(1 -φ K-1 )(1 -φ) (1 -φ) 2 - (K -1)φ K+1 -Kφ K + φ (1 -φ) 2 = φ (1 + 2φ) + (K -1) -(K -1)φ -φ 2 + φ K+1 (1 -φ) 2 = φ (1 + 2φ) + (K -1)(1 -φ) -φ 2 (1 -φ K-1 ) (1 -φ) 2 . ( 22
)
Subtracting this last expression to the one obtained for [π K-1,0 ] -1 we get the expression of λIE[T P ] :

λIE[T P ] = [1 + (K -1)(1 + ρ)] (1 -ρ)(1 + 2φ) = ρ (1 -ρ) (K -1) + 1 (1 + ρ) , (23)
and then the expression of the expectation scaled in term of a number of mean service time :

µIE[T P ] = 1 2(1 -ρ) (K -1) + 1 (1 + ρ) . ( 24
)
Note that µIE[T P ] represents also the expected number of custumers served by the intermittent server during a passage in the front office.

In Figure 5, we have plotted the scaled expectation of the pseudo-busy period of the second server as a function of ρ, for different values of the integer K. As we would expect, this expectation is increasing with ρ and with K. Moreover, we can say that the expectation of the pseudo-busy period of the second server is relatively small when ρ is between 0 and around 0.4, when we compare it with the one of the pseudo-idle period (cf. Figure 4). This shows the benefit of the intermittent server since the use of a low percentage of his time significantly decreases the mean waiting time.

Cost Function

We have to consider two somewhat different situations. The first one is when the second server is not necessary for the system to be stable (i.e., when ρ < 0.5). The second situation is when the second server is necessary to the system (ρ ≥ 0.5).

In the first situation, the second server just helps to decrease the mean waiting time IE[W ] seen by the customers. We have to compare this help to the customers with respect to the perturbation of the work done in the back office.

We assume here that there is a fixed penalty c 0 to pay each time the second server has to leave the back office and that the cost per unit of time of this second server is c 1 . We also assume that c 2 is the cost per unit of waiting time. During a unit time, the expectation of the cumulative value of the waiting times equals λIE[W ]; this expectation being nothing else than the expectation of the number of waiting

IE[N w ] = IE[N ] -2ρ . (25)
Then, depending on the value K, the function to minimize corresponds to the expected total variable cost per time unit, and is given by :

C(K) = c 0 [IE[T A ] + IE[T P ]] -1 + c 1 S 1 + c 2 IE[N w ] , (26)
where here also, S 1 denotes the sum

K-1 i=1 π i,1 + ∞ i=K π i .
Note that this sum of probabilities S 1 is nothing but the mean time per time unit spent by the second server in the front office.

When the variable K is increased, the first two terms are decreasing while the term c 2 IE[N w ] is increasing. More precisely, considering a cycle of the intermittent server, we start from the relation :

S 1 = IE[T P ] (IE[T A ] + IE[T P ]) = 1 1 + IE[T A ] IE[T P ] . (27)
Considering equations ( 22) and ( 23) we deduce that, when K tends to infinity, the two expectations tend to infinity. Considering now the ratio

IE[T A ] IE[T P ]
when K tends to infinity, since φ satisfies φ > 1, the limit of this ratio is the same as the limit of the following ratio :

lim K-→+∞ IE[T A ] IE[T P ] = lim K-→+∞ (2φ -1) (1 -φ) 2 φ K K = +∞ . (28)
Therefore, the first two terms of the cost function tends asymptotically to zero when K tends to infinity while the term c 2 IE[N w ] is increasing (from c 2 2ρ 3 /(1 -ρ 2 ) when K = 2 to the asymptotic value c 2 4ρ 2 /(1 -2ρ) when K tends to infinity). In this situation The optimal K may not be finite if the penalty coefficient c 2 is not large enough.

The second situation is different in the sense that K has to be finite in order to have a stable solution.

In this case, the intermittent server has to work in the front office a percentage of time S 1 greater than (λ/µ -1) in order that the system admits a steady state solution. The maximal feasible value K max of K is given by K max = max{K|S 1 (K) > λ/µ -1}. Practically, if K max is large enough (i.e., when (λ/µ -1) is not close to unity), the cost c 2 IE[N w ] should be large when K = K max and we may expect the cost function to be convex. However, the convexity of C(K) has not been investigated theoretically. Also, from a practical point of view, the parameter c 2 has again to be not too small with respect to c 0 and c 1 in order to avoid the limit behavior where the second server would come once a year to empty the waiting room. In Figure 6, we have plotted two sets of values of C(K) when ρ = 0.35 (alone the permanent server queue would have a utilization factor of 0.7), for c 0 = 0.5, c 1 = 1. Case c 2 = 0.15 (noted with stars) gives an optimal K * = 5. From Figures (4) and ( 5), we can check that for this optimal solution the mean pseudo-idle period of the second server is around 70 times the mean service time while the mean pseudo-busy period is close to 5 times the mean service time. But case c 2 = 0.03 (noted with sign +) gives a decreasing cost for K ∈ [2, 30].

Conclusions

We have shown in this paper the importance of intermittent servers in order to reduce the response times without increasing significantly the idle times of servers. For such situations where a single server would satisfy the stability condition (λ < µ), a non trivial result is that the pseudo-idle period of the second server is significantly longer than what would be generally expected by the management and also that the pseudo-busy period stays small; and so the second server can keep his main activity in the back office.

We can think of applications in architectures for quite large telecommunication switches where we have "guard" processors to help the congested input queues on demand. It may also help in the context of network function virtualization (NFV) in which a service might be deployed on demand to face a transient congestion. Not only these results are interesting by themselves if such a situation occurs in a real situation but also, this study can be used to check simulation models used for a more complex situation.

Appendix 1: Determination of eqn. (12)

Starting from the normalizing equation :

S 0 + S 1 = 1 , (29)
where

S 0 = π 0 + K-1 i=1 π i,0 , and S 1 = π 0 + K-1 i=1 π i,1 + ∞ i=K π i ,
we first consider the partial sum S 0 :

S 0 = π 0 + K-1 i=1 π i,0 = π 0 + K-1 i=1 π K-i,0 , = π 0 + π K-1,0 K-1 i=1 ( i-1 j=0 φ j ) = π 0 + π K-1,0 K-1 i=1 (K -i)φ i-1 , = π 0 + π K-1,0 K K-1 i=1 φ i-1 - K-1 i=1 iφ i-1 ,
or ( if φ = 1) :

S 0 = π 0 + π K-1,0 K (1 -φ K-1 ) (1 -φ) - (1 -Kφ K-1 + (K -1)φ K ) (1 -φ) 2 , = π 0 + π K-1,0 (1 -φ) K(1 -φ) -(1 -φ K ) (1 -φ) , = π 0 + π 0 (1 + 2φ) D 0 K(1 -φ) -(1 -φ K ) (1 -φ) = π 0 1 + (1 + 2φ)(K(1 -φ) -(1 -φ K )) (1 -φ)D 0 , = π 0 (1 -φ)D 0 φ(1 -φ) 2 + (1 + 2φ)[(K -1)(1 -φ) -φ 2 (1 -φ K-1 )] .
Considering now the partial sum S 1 , i.e., the steady state probability that the back-office server is helping the front-office server, we have :

S 1 = K-1 i=1 π i,1 + ∞ i=K π i , = π 1,1 K-1 i=1 (1 + ρ) -2ρ i (1 -ρ) + π K ∞ i=K ρ i-K , = π 1,1 (K -1)(1 + ρ) (1 -ρ) -2π 1,1 1 (1 -ρ) K-1 i=1 ρ i + π K 1 (1 -ρ) , = π 1,1 (K -1)(1 + ρ) (1 -ρ) -π 1,1 2 (1 -ρ) (1 -ρ K ) (1 -ρ) + π 1,1 (1 + ρ) -2ρ K (1 -ρ) 1 (1 -ρ) , = π 1,1 (K -1)(1 + ρ) (1 -ρ) -π 1,1 1 (1 -ρ) = π 1,1 1 + (K -1)(1 + ρ) (1 -ρ) , = π 0 (1 -φ)[K + ρ(K -1)] (1 -ρ)D 0 .
Using the normalizing equation, i.e., S 0 + S 1 = 1, we get the expression of probability π 0 when φ = 1 :

π 0 = (1 -ρ)(1 -φ)D 0 D 1 , (30)
where

D 1 = (1 -ρ){φ(1 -φ) 2 + (1 + 2φ)[(K -1)(1 -φ) -φ 2 (1 -φ K-1 )]} + +(1 -φ) 2 [K + ρ(K -1)] .
For the special case where φ = 1, it is not difficult, starting from the specific relations between probabilities given at the end of Section 3.1, to find the following expressions :

S 0 = 3K(K + 1) -4 2(3K -2) π 0 , S 1 = (3K -1) (3K -2) π 0 , π 0 = 2(3K -2) 3(K(K + 3) -2) . ( 31
)
8 Appendix 2: Determination of mean number of customers

In order to obtain the expression, let us start by computing two partial sums (B 0 and B 1 ), under the condition φ = 1 :

B 0 = K-1 i=1 iπ i,0 = K-1 i=1 (K -i)π K-i,0 = π K-1,0 K-1 i=1 (K -i)( i-1 j=0 φ j ) , = π K-1,0 K-1 i=1 (K -i) (1 -φ i ) (1 -φ) = π K-1,0 (1 -φ) K-1 i=1 i -K K-1 i=0 φ i + K + φ K-1 i=1 iφ i-1 , = π K-1,0 (1 -φ)
K(K + 1) 2 -K K-1 i=0 φ i + φ K-1 i=1 iφ i-1 , = π K-1,0 (1 -φ) K(K + 1) 2 - K (1 -φ) + φ(1 -φ K ) (1 -φ) 2 , = π 0 (1 + 2φ) D 0 K(K + 1) 2 - K (1 -φ) + φ(1 -φ K ) (1 -φ) 2 , = (1 -ρ)(1 -φ)(1 + 2φ) D 1 K(K + 1) 2 - K (1 -φ) + φ(1 -φ K ) (1 -φ) 2 ,
and secondly :

B 1 = K-1 i=1 iπ i,1 + ∞ i=K iπ i = K-1 i=1 iπ i,1 + π K ∞ i=K iρ i-K , = π 1,1 K-1 i=1 i (1 + ρ) (1 -ρ) - K-1 i=1 2iρ i (1 -ρ) + π 1,1 (1 + ρ) -2ρ K (1 -ρ) ∞ i=K iρ i-K , = π 1,1 (1 + ρ) (1 -ρ) K-1 i=1 i - 2π 1,1 (1 -ρ) ∞ i=1 iρ i + π 1,1 (1 + ρ) (1 -ρ) ∞ i=0 (K + i)ρ i , = π 1,1 (1 -ρ) (1 + ρ) K(K -1) 2 - ρ (1 -ρ) + K (1 + ρ) (1 -ρ) , = π 0 (1 -φ) (1 -ρ)D 0 (1 + ρ) K(K -1) 2 + K + ρ(K -1) (1 -ρ) = (1 -φ) 2 D 1 (1 + ρ) K(K -1) 2 + K + ρ(K -1) (1 -ρ) .
From that we get the expression of the expectation of the number of customers :

IE[N ] = K-1 i=1 iπ i,0 + K-1 i=1 iπ i,1 + ∞ i=K iπ i , = (1 -ρ)(1 -φ)(1 + 2φ) D 1 K(K + 1) 2 - (K) (1 -φ) + φ(1 -φ K ) (1 -φ) 2 + + (1 -φ) 2 D 1 (1 + ρ) K(K -1) 2 + K + ρ(K -1) (1 -ρ) , = (1 -φ) D 1 (1 -ρ)(1 + 2φ) K(K + 1) 2 - K (1 -φ) + φ(1 -φ K ) (1 -φ) 2 + +(1 -φ) (1 + ρ) K(K -1) 2 + K + ρ(K -1) (1 -ρ) .
This last result corresponds to the expression presented in Section 3.2.

In the special situation where φ = 1, let us first consider the sum B 0 . Starting from the equality obtained above

B 0 = π K-1,0 K-1 i=1 (K -i)( i-1 j=0 φ j ),
we get :

B 0 = π K-1,0 K-1 i=1 (K -i)i , = π K-1,0 K K-1 i=1 i - K-1 i=1 i 2 = π K-1,0 K K(K + 1) 2 - (K -1)K(2K -1) 6 ,
= π K-1,0 (K -1)K(K + 1) 6 = π 0 3 3K -2 (K -1)K(K + 1) 6 , = π 0 (K -1)K(K + 1) 2(3K -2) .

Let us now consider the sum B 1 . We may start from the following equality obtained above

B 1 = π 1,1 (1 -ρ) (1 + ρ) K(K -1) 2 - ρ (1 -ρ) + K (1 + ρ) (1 -ρ) ,
and since here ρ = 1/2, we get :

B 1 = π 1,1 2
(3K(K + 3) -4) = π 0 2(3K -2) (3K(K + 3) -4) .

After summation of B 0 and B 1 and use of the expression of π 0 given by relation 31, we are able to exhibit the following expression :

IE[N ] = K(K(K + 3) + 8) -4 (K(K + 3) -2)
.

Figure 1 :

 1 Figure 1: Transition graph of the CTMC.

Figure 2 :

 2 Figure 2: Transition graph of the CTMC when K = 2.

Figure 3 :

 3 Figure 3: Mean number of customers as a function of ρ. For (bottom-up) K = 2, 3, 4, 5 and 6. Curve with stars corresponds to infinite K, the second server being never called.

Figure 4 :

 4 Figure 4: Scaled expectation of the pseudo-idle period of the second server as a function of ρ. For (bottom-up) K = 2, 3, 4, 5 and 6.

Figure 5 :

 5 Figure 5: Scaled expectation of the pseudo-busy period of the second server as a function of ρ. For (bottom-up) K = 2, 3, 4, 5 and 6.

Figure 6 :

 6 Figure 6: Variable cost function, with ρ = 0.35, c 0 = 0.5, c 1 = 1. Case 1 (stars): c 2 = 0.15. Case 2 (sign +): c 2 = 0.03.

  Given that the CTMC is in state (2, 1), if a service completes before a new arrival, the CTMC joins either state (1, 0) if the second server finishes his service first or state (1, 1) in the other case. These two events have equal probabilities (0.5 each). If the CTMC joins state (1, 1) from state (2, 1), this means that , given a service completes when the CTMC is in state (2, 1), the CTMC goes to state (1, 0) with probability 0.5, goes to state (0) without coming back to state (2, 1) with probability 0.5 ×

	Soµ λ + µ λ comes back to state (2, 1) with probability 0.5 × λ + µ	or

the permanent server becomes idle. Then either the second server becomes idle (with probability µ λ + µ )

or the regular server becomes busy again, the CTMC revisiting state (2, 1) (with probability λ λ + µ

).


Clément Livache

Bertille Martinez

Nicolas Goubet

Julien Ramade

Emmanuel Lhuillier

Road Map for Nanocrystal based Infrared Photodetectors

Keywords: Infrared, nanocrystals, photodetection, device, interband, intraband

come

INTRODUCTION

Over the recent years, colloidal quantum dots (CQDs) have reached a first mass market application with their use as light sources for displays. This application brought even more interest for CQDs both at the academic and industrial levels. Among emerging applications, infrared (IR) photodetection [START_REF] Kershaw | Narrow bandgap colloidal metal chalcogenide quantum dots: synthetic methods, heterostructures, assemblies, electronic and infrared optical properties[END_REF][START_REF] Lhuillier | Recent Progresses in Mid Infrared Nanocrystal Optoelectronics[END_REF]) is a field where colloidal materials have a strong potential to bring cost disruption, especially because organic semiconductors, often seen as the low-cost alternative to conventional semiconductors, are ineffective in this range of wavelengths.

IR detection currently relies on two types of sensors. Quantum detectors are based on photon-absorbing semiconductor materials. These can either be narrow band gap semiconductors (InGaAs in the short-wave IR, InSb in the mid-wave and HgCdTe for both mid-and long-wave IR) or semiconductor heterostructures (GaAs/AlGaAs, used in Quantum Well Infrared Photodetector or QWIP, and InAs/GaSb in type II superlattices). These technologies are mature, present high performances (i.e. high quantum efficiency, relatively low dark current, high uniformity, fast time response), but suffer from an excessive cost (a typical IR camera costs 30k-100 k€) and a low operating temperature. As a result, access to this type of technologies remains restricted to defense and scientific applications (mostly astronomy).

The second class of IR detectors is thermal detectors, sensitive to energy flux rather than photon flux. Materials used for this kind of detectors see one of their physical property (typically, their electrical resistance) change upon absorption of IR radiation. Typical technologies are bolometers and pyrometers. Their operation principles make them intrinsically slower than quantum devices, and they present lower detectivity (signal to noise ratio) than their quantum detector counterpart. On the other hand, they can be operated at room temperature and their cost is significantly lower, ranging from 100 € to few k€ per focal plane array (FPA) and 1k € to 10 k€ for camera.

To bring IR detection to a mass market level, a technology combining both the performances of quantum detector and the low cost of thermal detector needs to emerge. CQDs appear as promising candidates to reach this goal. Beyond their tunable absorption from the near IR to the THz range (Goubet et al., 2018a), several significant proofs of concepts such as mid-IR photoconduction [START_REF] Keuleyan | Mid-infrared HgTe colloidal quantum dot photodetectors[END_REF], BLIP limited photodiode [START_REF] Guyot-Sionnest | Background limited mid-infrared photodetection with photovoltaic HgTe colloidal quantum dots[END_REF], investigation of stability issues (Jagtap et al., 2018a) have brought CQDs to a technological readiness level (TRL) above 3 which is critical for the industry to start considering an emerging technology.

The focus of this paper is intentionally limited to IR detection, which means that we have excluded from the scope of this review any solar cell application [START_REF] Sargent | Colloidal quantum dot solar cells[END_REF]. In this perspective, we propose a road map of the main challenges that have to be addressed by the community in order to transfer the IR CQD technology to the industrial level. We have sorted these challenges into three main categories: (i) material, (ii) device and (iii) camera integration challenges, see Figure 2.

DISCUSSION

Basic of IR detection using nanocrystal as active material

To start, we would like to discuss the basics of the transformation of a colloidal nanocrystal solution into an IR sensor. Two type of geometries have been explored: planar and vertical geometry, see Figure 1a andc. The planar geometry is certainly the easiest to implement, because this geometry is far less sensitive to the film quality (ie film roughness and cracks do not lead to electrical shorts in the device). The success of this geometry also relates to the possibility to add a gate for the design of field effect (photo)transistor [START_REF] Talapin | PbSe Nanocrystal Solids for n-and p-Channel Thin Film Field-Effect Transistors[END_REF]. Typically, electrodes are prepared on a conventional substrate (Si/SiO2 typically). Interdigitated electrodes have been widely used as a strategy to enhance the current magnitude. The film of CQD is deposited on this substrate using methods such as spin coating, dropcasting, dip coating or spray coating [START_REF] Cryer | 300 nm Spectral Resolution in the Mid-Infrared with Robust, High Responsivity Flexible Colloidal Quantum Dot Devices at Room Temperature[END_REF]. As is, the film of nanocrystal is insulating and a ligand exchange step is necessary to increase the CQD electronic coupling and achieve photoconduction. IR exposition is obtained by top side illumination. Typical I-V curve from such planar photoconductive device is shown in Figure 1b. The photosignal relates to the modulation of the I-V curve slope. Field effect configuration (Lhuillier et al., 2014b) is interesting to tune the majority carrier current and possibly enhance the signal to noise ratio. The gating is typically obtained through the use of the dielectric layer from the substrate or through the deposition of a top side dielectric [START_REF] Chung | Low Voltage, Hysteresis Free, and High Mobility Transistors from All-Inorganic Colloidal Nanocrystals[END_REF] or electrolyte (Lhuillier et al., 2014b). The second type of detector geometry that has been widely explored is based on a vertical geometry. The typical stack of layers relies on a transparent substrate (glass in the near infrared) on top of which a transparent conductive layer is deposited. ITO (indium tin oxide) and FTO (fluoride doped tin oxide) are the most used material. An electron transport layer (ETL) is then deposited: the most used material for CQD based device are inorganic layer made of ZnO and TiO2. This layer generally needs to be annealed at high temperature, which make that it is highly desirable to process it as a bottom layer (ie before the CQD deposition). On the top of the ETL, the CQD layer is deposited. The typical thickness of this layer is 200 to 400 nm. This value is a trade-off: thicker layer might be desirable to absorb more light since only 10-30 % of the light are absorbed in those conditions [START_REF] Cademartiri | Size-Dependent Extinction Coefficients of PbS Quantum Dots[END_REF][START_REF] Hens | Light absorption by colloidal semiconductor quantum dots[END_REF]. However, thicker layers are difficult to build due to the multiplication of deposition and ligand exchange steps. Moreover, the short transport diffusion length makes that photocarriers might not be collected in thicker layers. On the top of the QD layer, a hole transport layer can be deposited. MoO3 has been intensively used in the case of CQD [START_REF] Chuang | Improved performance and stability in quantum dot solar cells through band alignment engineering[END_REF][START_REF] Gao | n-Type Transition Metal Oxide as a Hole Extraction Layer in PbS Quantum Dot Solar Cells[END_REF]. Finally, a top metallic contact is deposited. There are many possible alternative configurations to the one described above with inverted geometry, as the combination of a n and p type layer [START_REF] Chuang | Improved performance and stability in quantum dot solar cells through band alignment engineering[END_REF] or graded band gap configuration [START_REF] Kramer | Solar Cells Using Quantum Funnels[END_REF] to funnel charges to the contacts. In this vertical configuration, illumination is made though the substrate, bottom contact and ETL. A typical IV curve of such photodiode is shown in Figure 1d. The key advantage of this configuration is to be able to operate the device close to zero bias to reduce the dark current, by taking advantage of the built-in electric field of the diode. Now that the basic of CQD baser IR detector design is established, it is of utmost importance to remind the main figures of merit relative to IR sensing [START_REF] Rosencher | Optoelectronics[END_REF]. Responsivity (in A/W) is the first figure of merit which translates the ability of the active layer to transform a light signal into an electrical signal. This quantity directly relates to the external quantum efficiency (efficiency to convert incident photons into electrical current) and to internal quantum efficiency (efficiency to convert absorbed photons into electrical current, in other word the external quantum efficiency normalized by the device absorption). One of the key specificities of IR is the limited signal-to-noise ratio. Indeed, because of the narrow energy transition involved in the IR, thermal activation competes with photon activation of the carriers. This results in a dark current which can be a significant fraction of the total current. The relevant contribution of the dark current to noise is its spectral distribution (in A.Hz -1/2 ), hence the quantity involved in the ultimate figure of merit of an IR detector is the detectivity (signal to noise ratio expressed in cm.Hz 1/2 .W -1 or Jones). Currently, all convincing reports relative to the measurement of noise in nanocrystal arrays have led to 1/f noise as the prevailing contribution [START_REF] De Iacovo | Noise performance of PbS colloidal quantum dot photodetectors[END_REF][START_REF] Lai | Low-Frequency (1/f) Noise in Nanocrystal Field-Effect Transistors[END_REF][START_REF] Liu | 1/f noise in semiconductor and metal nanocrystal solids[END_REF]. It is a very common habit to observe detectivity value reported assuming that noise is shot noise limited (mostly because there is an analytical expression for shot noise and none for 1/f noise), however this leads to a huge overestimation of the device detectivity. Finally, another important figure of merit which differentiates detectors from solar cells is the device time response. To take full advantage of photon detectors, faster time responses that the ones reported for thermal detectors (≈10 ms) are highly desirable. In the following we discuss state of the art results and expected performance targets for SWIR, MWIR and LWIR range of wavelengths.

Material challenges

IR absorption: interband vs intraband transitions

Among all criteria to build an IR detector, design IR absorption appears as the first challenge to tackle. Three wavelength bands appear promising for applications: short-, mid-and long-wave infrared.

Short-Wave Infrared (SWIR) extends from 800 nm to 1.7 µm, and up to 2.5 µm for extended SWIR. In this range, the objective for CQD-based detectors is to offer an alternative to InGaAs. As stated earlier, these technologies offer toplevel performances. However, their cost, without being prohibitive, remains far above comparable technologies in the visible range. Moreover, the perspectives of cost disruption are limited for such a mature technology. In this range of wavelengths, applications are typically active imaging, night glow assisted imaging and tissues imaging. Among possible colloidal materials to be used in this range of wavelengths, two materials have reached a large enough maturity : lead chalcogenides [START_REF] Sargent | Solar Cells, Photodetectors, and Optical Sources from Infrared Colloidal Quantum Dots[END_REF] , 20) (PbS andPbSe, mostly) andHgTe (Green andMirzai, 2018; Keuleyan et al., 2011;[START_REF] Kovalenko | Colloidal HgTe Nanocrystals with Widely Tunable Narrow Band Gap Energies: From Telecommunications to Molecular Vibrations[END_REF]. In those materials, IR absorption is obtained through interband transitions, see Figure 3a.

In the Mid-Wave Infrared (MWIR: 3-5 µm), blackbody emission of room-temperature objects starts to prevail over reflection of other light sources, hence opening the field of thermal imaging. In this range, HgTe is by far the most investigated material [START_REF] Keuleyan | Mid-infrared HgTe colloidal quantum dot photodetectors[END_REF][START_REF] Kovalenko | Colloidal HgTe Nanocrystals with Widely Tunable Narrow Band Gap Energies: From Telecommunications to Molecular Vibrations[END_REF][START_REF] Tang | Scalable Fabrication of Infrared Detectors with Multispectral Photoresponse Based on Patterned Colloidal Quantum Dot Films[END_REF] thanks to its tunable interband IR transition, see Figure 3d. Another strategy to achieve low-energy transition in the MWIR is to use intraband transitions (see Figure 3b) [START_REF] Deng | Colloidal Quantum Dots Intraband Photodetectors[END_REF]Jagtap et al., 2018b;[START_REF] Kim | Self-doped colloidal semiconductor nanocrystals with intraband transitions in steady state[END_REF]. In this case, the transition occurs in the first levels of the conduction band, hence doped semiconductors are necessary. Again, mercury chalcogenides are the most investigated materials for photodetection thanks to self-doping [START_REF] Deng | Colloidal Quantum Dots Intraband Photodetectors[END_REF]Jagtap et al., 2018b;[START_REF] Kim | Self-doped colloidal semiconductor nanocrystals with intraband transitions in steady state[END_REF].

Long-Wave Infrared (LWIR: 5-30 µm) is the optimal range to perform thermal imaging of room-temperature objects since their blackbody emission maximum lies around 10 µm. Addressing such low energy with interband transitions is extremely challenging since the confinement energy needs to be so small that the required size of nanocrystals becomes incompatible with colloidal stability or monodispersity. Intraband [START_REF] Park | Mid-Infrared Intraband Transition of Metal Excess Colloidal Ag2Se Nanocrystals[END_REF] and plasmonic transitions [START_REF] Agrawal | Control of Localized Surface Plasmon Resonances in Metal Oxide Nanocrystals[END_REF][START_REF] Askari | Low-Loss and Tunable Localized Mid-Infrared Plasmons in Nanocrystals of Highly Degenerate InN[END_REF][START_REF] Coughlan | Compound Copper Chalcogenide Nanocrystals[END_REF][START_REF] Liu | Charge Storage and Quantum Confinement Resilience in Colloidal Indium Nitride Nanocrystals[END_REF][START_REF] Luther | Localized surface plasmon resonances arising from free carriers in doped quantum dots[END_REF]) (achieved at a higher doping level) in doped nanocrystals, see Figure 3c, are interesting for two reasons: (i) addressing long wavelengths from the MWIR to the THz range while keeping the colloidal stability of the material and (ii) because doping of nanocrystals might not be limited to toxic material.

Surface chemistry

Beyond the nanocrystal synthesis, the control of surface chemistry is a critical step toward the design of photoconductive thin films. Initial long capping ligands, which ensure the nanometer size growth and preserve the colloidal stability, need to be stripped from the surface to increase the inter-CQD coupling (ie to reduce the inter CQD tunnel barrier) and obtain reasonable carrier mobility. For most of the reported devices the ligand exchange remains based on a solid state ligand exchange (ie performed on the film), typically using ethanedithiol as capping ligand [START_REF] Lhuillier | Mid-Infrared HgTe/As2S3 Field Effect Transistors and Photodetectors[END_REF]. This undoubtedly limits the carrier mobility in the 10 -3 cm 2 V -1 s -1 range and likely the associated photoresponse. Introduction of inorganic ligands such as As2S3 appears as an interesting path to obtain higher film mobility [START_REF] Cryer | 300 nm Spectral Resolution in the Mid-Infrared with Robust, High Responsivity Flexible Colloidal Quantum Dot Devices at Room Temperature[END_REF][START_REF] Lhuillier | Mid-Infrared HgTe/As2S3 Field Effect Transistors and Photodetectors[END_REF][START_REF] Tang | Scalable Fabrication of Infrared Detectors with Multispectral Photoresponse Based on Patterned Colloidal Quantum Dot Films[END_REF][START_REF] Yakunin | High Infrared Photoconductivity in Films of Arsenic-Sulfide-Encapsulated Lead-Sulfide Nanocrystals[END_REF]. It is nevertheless as important to boost mobility as preserving a good CQD surface passivation, especially for the design of photodiode, and more work needs to be done in this direction. In the case of ink preparation, CQD ends up being ligand-exchanged and suspended in polar solvent with high boiling point, raising some questions relative to the film preparation. Indeed, most of the devices are currently prepared using dropcasting [START_REF] Tang | Thermal Imaging with Plasmon Resonance Enhanced HgTe Colloidal Quantum Dot Photovoltaic Devices[END_REF] or spin coating. The latter method is difficult to implement using high boiling point solvent, in addition to a dramatically low efficiency (ie 90% of the material is wasted). Among alternative methods, dip coating [START_REF] Chernomordik | Quantum Dot Solar Cell Fabrication Protocols[END_REF] and spray coating [START_REF] Chen | Fast, Air-Stable Infrared Photodetectors based on Spray-Deposited Aqueous HgTe Quantum Dots[END_REF][START_REF] Cryer | 300 nm Spectral Resolution in the Mid-Infrared with Robust, High Responsivity Flexible Colloidal Quantum Dot Devices at Room Temperature[END_REF][START_REF] Wang | Effects of electron-phonon interactions on the electron tunneling spectrum of PbS quantum dots[END_REF] have also been reported. The choice for a given method also impacts the preparation of the QD ink: low concentrations are used for dropcasting (10 mg/mL), while higher concentrations (50 mg/mL) are used for spin coating in non-polar solvent and even higher in the case of spin coating from polar solvent. There is probably no perfect method for deposition and each involved team has to pay the price of time-consuming optimization for this step.

Another main challenge relative to the use of mercury chalcogenides relates to the softness of the material. As a result, any annealing step (to boost the mobility or as part of a lithography process) leads to sintering of the nanocrystal film. This induces an increase of the CQD effective radius which broadens the absorption cut-off and, even worse, dramatically increases the dark current (through a reduction of the effective band gap). Thus, core shell objects with an external material which is able to sustain temperatures around 160°C (ie typical baking temperature of lithography resists such as Polymethyl methacrylate) without aggregation will be of utmost interest. Introduction of core shell structure may also lead to longer living photocarriers which is highly desirable for photodetection. A first study have been done in this direction by coupling HgTe and HgSe materials in heterostructured nanocrystals (Goubet et al., 2018b). However, growing such shell remains quite challenging because HgX compounds are grown at low temperature (100 °C typically), while conventional shell materials are synthetized at high temperature (200 °C and more). Currently all reported HgX based core-shell materials remain based on the room temperature Colloidal-Atomic Layer Deposition (C-ALD) process [START_REF] Robin | Surface Control of Doping in Self-Doped Nanocrystals[END_REF][START_REF] Sagar | HgSe/CdE (E = S, Se) Core/Shell Nanocrystals by Colloidal Atomic Layer Deposition[END_REF][START_REF] Shen | HgS and HgS/CdS Colloidal Quantum Dots with Infrared Intraband Transitions and Emergence of a Surface Plasmon[END_REF]), which appears to be suboptimal procedure (Lhuillier et al., 2014a). Shell growth method specific to soft materials will have to be developed.

Material toxicity

Material toxicity is probably one of the most difficult issues to address. As stated previously, the field remains driven by lead and mercury chalcogenides, which are actually the same material as the ones used in current IR detection technologies. For sure, the introduction of low toxicity compounds will be a breakthrough in the field. In the visible range of wavelengths, a large effort has been devoted to the synthesis of InP as an alternative to CdSe as light source for display. On the other hand, colloidal narrow band gap III-V materials (InAs [START_REF] Franke | Continuous injection synthesis of indium arsenide quantum dots emissive in the short-wavelength infrared[END_REF][START_REF] Grigel | InAs Colloidal Quantum Dots Synthesis via Aminopnictogen Precursor Chemistry[END_REF][START_REF] Srivastava | Monodisperse InAs Quantum Dots from Aminoarsine Precursors: Understanding the Role of Reducing Agent[END_REF] and InSb [START_REF] Chang | Carrier Dynamics in Highly Quantum-Confined, Colloidal Indium Antimonide Nanocrystals[END_REF][START_REF] Maurice | Quantum Dots: Synthesis of Colloidal Indium Antimonide Nanocrystals Using Stibine[END_REF]) remain poorly mature with a too limited amount of reports and none dedicated to IR detection further than the SWIR. First issue is due to the more covalent character of III-V materials with respect to II-VI materials. As a result, more reactive precursors need to be used which leads to a higher degree of synthesis complexity. Second, the availability of pnictogen precursors is decreasing for narrow band gap materials mostly due to their atomic radius increasing.

Narrow band gap interband transitions are more likely to be observed using heavy elements (higher Z value comes with a denser density of states, which are more likely to present a narrow band gap in the vicinity of the Fermi level). However, materials in the bottom part of the periodic table are also toxic. Certainly, a switch toward intraband transition is the promising path to achieve heavy metal free IR detection. This strategy is nevertheless currently limited to the MWIR and LWIR and the level of performance is by far not as good. Ag2Se, thanks to its spectral vicinity with HgSe has recently generated some interest for MWIR detection [START_REF] Park | Mid-Infrared Intraband Transition of Metal Excess Colloidal Ag2Se Nanocrystals[END_REF][START_REF] Qu | Intraband Mid-Infrared Transitions in Ag2Se Nanocrystals: Potential and Limitations for Hg-Free Low-Cost Photodetection[END_REF][START_REF] Sahu | Quantum confinement in silver selenide semiconductor nanocrystals[END_REF], although detection performances remain for now several orders of magnitude below than what have been reported with HgX compounds.

Another class of material that may appears as promising to achieve infrared absorption are strongly-doped semiconductors with plasmonic absorption in the infrared. This is typically the case of copper chalcogenides [START_REF] Coughlan | Compound Copper Chalcogenide Nanocrystals[END_REF][START_REF] Dorfs | Reversible Tunability of the Near-Infrared Valence Band Plasmon Resonance in Cu2-xSe Nanocrystals[END_REF][START_REF] Kriegel | Tuning the Excitonic and Plasmonic Properties of Copper Chalcogenide Nanocrystals[END_REF] , oxide nanoparticles [START_REF] Buonsanti | Tunable Infrared Absorption and Visible Transparency of Colloidal Aluminum-Doped Zinc Oxide Nanocrystals[END_REF][START_REF] Gaspera | Low-Temperature Processed Ga-Doped ZnO Coatings from Colloidal Inks[END_REF][START_REF] Ghosh | Tunable surface plasmon resonance and enhanced electrical conductivity of In doped ZnO colloidal nanocrystals[END_REF][START_REF] Kanehara | Indium Tin Oxide Nanoparticles with Compositionally Tunable Surface Plasmon Resonance Frequencies in the Near-IR Region[END_REF][START_REF] Runnerstrom | Defect Engineering in Plasmonic Metal Oxide Nanocrystals[END_REF][START_REF] Schimpf | Redox Chemistries and Plasmon Energies of Photodoped In2O3 and Sn-Doped In2O3 (ITO) Nanocrystals[END_REF][START_REF] Tandon | Size-Induced Enhancement of Carrier Density, LSPR Quality Factor, and Carrier Mobility in Cr-Sn Doped In2O3 Nanocrystals[END_REF] or doped silicon [START_REF] Gresback | Controlled Doping of Silicon Nanocrystals Investigated by Solution-Processed Field Effect Transistors[END_REF][START_REF] Zhang | Doped Silicon Nanocrystal Plasmonics[END_REF]. Plasmonic nanoparticles tend to have a much higher cross section (≈10 -13 cm 2 /particle) that the one associated with interband or intraband transitions (10 -15 -10 -14 cm 2 /particle range). However, their very short living photocarrier (<1 ps) might strongly balance the absorption enhancement. The investigation of (photo)transport properties in those materials needs to be pushed further.

Scale-up for production

When it comes to mass market application, a first question to consider is how much material will be necessary. Let's assume that the objective is to provide to every car sold in Europe (≈20 million units/per year) an IR sensor dedicated to night-driving assistance. Typical device will have a 1 cm 2 size for a thickness of a few hundreds of nm (400 nm for the calculation). Let's assume further a density of 10 for the material with a film filling of 0.64, corresponding to a randomly close-packed film. Moreover, we can account for the poor efficiency of deposition method such as spin coating where 90% of the material is wasted. A single device thus requires 2.6 mg of active material. This means that around 50 kg of HgTe will be necessary to saturate the targeted market. It is worth pointing that this amount is actually quite small and material supply is not an issue. This strongly contrasts with solar cell applications, where much larger devices (m 2 ) are necessary and where consequently material supplying (in particular Te, for which supply short fall is expected 10 years from now) and toxicity become critical issues.

Large scale synthesis of quantum dots [START_REF] Protière | Rational design of the gram-scale synthesis of nearly monodisperse semiconductor nanocrystals[END_REF] and more particularly mercury chalcogenides at the 10 g scale has already been reported (Lhuillier et al., 2016a). Thus, to reach an annual production of 50 kg, batches reaching a few hundred of g will be needed. This is probably a step where small and medium size companies producing nanocrystals should be involved to take advantage of the know-how developed for wide band gap materials (ie CdSe and InP).

Regarding the material production cost, it was recently evaluated by [START_REF] Jean | Synthesis cost dictates the commercial viability of lead sulfide and perovskite quantum dot photovoltaics[END_REF] that PbS CQDs present a fabrication cost in the 10 to 60 $/g. Thus, even assuming that for HgTe the fabrication cost will be in the upper range of this estimation, the cost of active material per device remains extremely low (0.15 $). We can conclude that contrary to solar cells, the material cost is here not a limiting factor.

Device challenges 3.1. Electronic structure and band alignment

As stated earlier, the most promising colloidal materials for IR detection are lead and mercury chalcogenides. Knowledge of the electronic structure of these material remains limited compared to the one reached for silicon and III-V semiconductors, and most electronic structure parameters are known within a limited accuracy, even for the bulk.

Once under colloidal form, we add on the top of that quantum confinement, dependence of the electronic spectrum with surface chemistry and surface traps. As a result, the electrical landscape of our active material remains quite blurry. It is consequently difficult to design device with carefully-optimized band alignments and ohmic contacts. A significant effort will have to be done to provide to the community data relative to the electronic structure of the infrared colloidal materials in a quite systematic way (material, size, surface chemistry). First results in this direction have been reported using combination of IR spectroscopy and electrochemistry (Chen and Guyot-Sionnest, 2017) or photoemission measurements [START_REF] Martinez | HgSe Self-Doped Nanocrystals as a Platform to Investigate the Effects of Vanishing Confinement[END_REF], but more will be needed.

As critical is the investigation of carrier dynamics in these narrow band gap materials especially for the understanding of current performances limitations. Methods based on time-resolved optical spectroscopy are difficult to transfer in the IR due to an intrinsically low PL efficiency of IR nanocrystals and due to less advanced optical setups in the IR. Development of alternative methods will be required. Again, some preliminary results in this direction have been obtained using time resolved photoluminescence (Keuleyan et al., 2014b), transient absorption [START_REF] Melnychuk | Slow Auger Relaxation in HgTe Colloidal Quantum Dots[END_REF], time resolved photoemission [START_REF] Livache | Charge Dynamics and Optolectronic Properties in HgTe Colloidal Quantum Wells[END_REF][START_REF] Spencer | Time-resolved surface photovoltage measurements at $n$-type photovoltaic surfaces: Si(111) and ZnO(10$\overline{1}$0)[END_REF] and transient photocurrent measurements [START_REF] Gao | Solution-Processed, High-Speed, and High-Quantum-Efficiency Quantum Dot Infrared Photodetectors[END_REF][START_REF] Livache | Band Edge Dynamics and Multiexciton Generation in Narrow Band Gap HgTe Nanocrystals[END_REF][START_REF] Martinez | Probing Charge Carrier Dynamics to Unveil the Role of Surface Ligands in HgTe Narrow Band Gap Nanocrystals[END_REF]. There are nevertheless not enough data to depict the full range of dynamics in this material from Auger recombination at short time scale to longlived traps at long time scale.

Thanks to the understanding of the electronic spectrum, the objective will be the design of new photodiodes, accounting for specificity of IR and colloidal material. While in the SWIR, exploiting the concept developed for solar cell was still a reasonable assumption (Jagtap et al., 2018a), this is no longer the case for longer wavelengths.

Alternatives to current electron (mostly ZnO and TiO2) and hole transport layers (MoO3) will have to be developed. This is even more true to implement the concept of unipolar barrier, which role is to let selectively one carrier flow, while preventing the other carrier to circulate. This concept has been widely used for III-V and II-VI semiconductor IR sensors [START_REF] Savich | Dark current filtering in unipolar barrier infrared detectors[END_REF][START_REF] Savich | Benefits and limitations of unipolar barriers in infrared photodetectors[END_REF][START_REF] White | Infra red detectors[END_REF], but remains poorly used in the case or CQD based devices. This might nevertheless be a time-consuming process because the fragility of the IR CQDs will likely require the development of specific carrier transport layer which doping and band alignment will have to be finely tuned and experimentally determined.

Device Performances

Since the main purpose of this road map is the design of effective IR-sensing devices, it is certainly worth determining which level of performances seems to be a reasonable goal to achieve for colloidal-based IR detectors in each specified range of wavelengths.

SWIR: For wavelengths up to 1.7 µm, the key competitor is the InGaAs technology which performance is again very unlikely to be beaten. Thus, CQDs appear promising for (i) cost disruption and (ii) extending the range of wavelengths toward the so-called extended SWIR (2.5 µm cut-off wavelength) [START_REF] Jia | Monolithic Integration of InSb Photodetector on Silicon for Mid-Infrared Silicon Photonics[END_REF]. InGaAs succeeded to achieve extremely low dark current (<20 fA at 20°C for a 15 µm pixel). Because CQD based devices reported so far are not reaching such low dark current densities, operational scenarii with high photon flux seems more appropriate. This typically relates to active imaging [START_REF] Geyer | Multispectral imaging via luminescent downshifting with colloidal quantum dots[END_REF], flame detection [START_REF] Iacovo | PbS Colloidal Quantum Dot Visible-Blind Photodetector for Early Indoor Fire Detection[END_REF] and biological tissues imaging for which reasonably fast detection is necessary (sub-ms is mandatory and µs is probably a more appropriate target) and already reported [START_REF] Lhuillier | Mid-Infrared HgTe/As2S3 Field Effect Transistors and Photodetectors[END_REF]. Also note that in the SWIR, room temperature operation or at least above water freezing point has to be achieved to preserve the low-cost character of the device. Because this range of wavelengths is fairly easy to reach using PbS CQDs and because of the relatively easy device characterization at such wavelengths, very high performance devices (up to kA.W -1 for responsivity and detectivity reaching 10 13 Jones at 1.4 µm) have been achieved [START_REF] Yakunin | High Infrared Photoconductivity in Films of Arsenic-Sulfide-Encapsulated Lead-Sulfide Nanocrystals[END_REF]. Device demonstration also includes imaging systems [START_REF] Calvez | Enabling NIR imaging at room temperature using quantum dots[END_REF][START_REF] Klem | Colloidal quantum dot Vis-SWIR imaging: demonstration of a focal plane array and camera prototype[END_REF].

MWIR: In this range, targeted applications are thermal imaging [START_REF] Tang | Thermal Imaging with Plasmon Resonance Enhanced HgTe Colloidal Quantum Dot Photovoltaic Devices[END_REF] and gas sensing (Chen et al., 2017). Current bulk-based imaging devices (typically based on InSb and HgCdTe) have presently no road map for operating temperature above 180 °C. This clearly sets a first objective. In the scope of preserving low-cost, it likely suggests that only Peltier cooling should actually be used, bringing the targeted operating temperature around 250 K. recently reported reduced Auger effect [START_REF] Melnychuk | Slow Auger Relaxation in HgTe Colloidal Quantum Dots[END_REF] in HgTe CQD compared to their bulk HgCdTe equivalent raises great hope to achieve this goal. In terms of performances, detectivity in the few 10 9 Jones at room temperature and reaching 10 10 Jones at 250 K is a clear objective which will bring CQD based technologies above what can be achieved using thermal sensors. Huge progresses in this direction have been reported this year. This includes responsivity above 1 A.W -1 [START_REF] Tang | Thermal Imaging with Plasmon Resonance Enhanced HgTe Colloidal Quantum Dot Photovoltaic Devices[END_REF], detectivity above 10 9 Jones at room temperature [START_REF] Cryer | 300 nm Spectral Resolution in the Mid-Infrared with Robust, High Responsivity Flexible Colloidal Quantum Dot Devices at Room Temperature[END_REF] and NETD (noise equivalent temperature difference) down to 14 mK [START_REF] Tang | Thermal Imaging with Plasmon Resonance Enhanced HgTe Colloidal Quantum Dot Photovoltaic Devices[END_REF]. In the MWIR, the device complexity lags far behind what has been achieved in the SWIR with just a few photodiodes [START_REF] Ackerman | Fast and Sensitive Colloidal Quantum Dot Mid-Wave Infrared Photodetectors[END_REF][START_REF] Guyot-Sionnest | Background limited mid-infrared photodetection with photovoltaic HgTe colloidal quantum dots[END_REF] and only one report for focal plane array integration [START_REF] Buurma | MWIR imaging with low cost colloidal quantum dot films[END_REF].

LWIR: When long wavelengths (8 µm and more) start being involved, the number of reported devices quickly drops (Keuleyan et al., 2014a). It consequently becomes difficult to set some objectives of performance. One key difficulty in this range of wavelengths comes from the fact that it is unlikely that non-cryogenic operating temperatures can be achieved. Thus, the cost disruption brought by the CQD is not as important as in the MWIR. Since the operating temperature of conventional IR camera in the 8-12 µm range is around 90 K, it is likely that operating temperature for CQD based device around 150 K will be desirable, with a detectivity remaining above 10 10 jones. Preserving a fast response is also key aspect to compete with bolometers. Regarding the THz range, there is currently no available data.

Device geometry and emerging strategies

Among current difficulties relative to colloidal materials, one key limitation remains connected to the short carrier diffusion length (50 to 100 nm typically) which remains shorter than typical absorption length (close to 10 µm [START_REF] Lhuillier | Optical properties of HgTe colloidal quantum dots[END_REF] for a band edge in the MWIR), due to the low carrier mobility. In other words, photogenerated carrier are only collected in the vicinity of the electrodes, while the bulk of the film leads to photogenerated charged that end up being trapped. In this sense, strategies to enhance the light-matter coupling are necessary. The objective of this strategy is to concentrate the incident electromagnetic field on the thin optically absorbing layer of QD. Some early results have been reported with the introduction of colloidal gold nanorods [START_REF] Chen | Photocurrent Enhancement of HgTe Quantum Dot Photodiodes by Plasmonic Gold Nanorod Structures[END_REF] or of resonators [START_REF] Tang | Thermal Imaging with Plasmon Resonance Enhanced HgTe Colloidal Quantum Dot Photovoltaic Devices[END_REF][START_REF] Yifat | Mid-IR colloidal quantum dot detectors enhanced by optical nano-antennas[END_REF]. The development of such plasmonic resonance has also been used to obtain polarized emission and imaging system (Le- [START_REF] Le-Van | Electrically driven optical metamaterials[END_REF][START_REF] Yifat | Mid-IR colloidal quantum dot detectors enhanced by optical nano-antennas[END_REF].

Another interesting development that has been reported relates to the design of multicolor detectors. This includes visible and MWIR (Lhuillier et al., 2014b), MWIR/LWIR [START_REF] Tang | Scalable Fabrication of Infrared Detectors with Multispectral Photoresponse Based on Patterned Colloidal Quantum Dot Films[END_REF] and MWIR/MWIR [START_REF] Cryer | 300 nm Spectral Resolution in the Mid-Infrared with Robust, High Responsivity Flexible Colloidal Quantum Dot Devices at Room Temperature[END_REF] sensors. However it remains unclear if bicolor technology can be compatible with low fabrication cost.

Finally, it is worth mentioning a strategy which has been explored over the last 5 years to boost the device photoresponse is the coupling of the nanocrystals with a 2D material. Graphene is the first to have been explored. Gigantic responsivities (10 7 A.W -1 ) have been reported [START_REF] Konstantatos | Hybrid graphene-quantum dot phototransistors with ultrahigh gain[END_REF][START_REF] Sun | Infrared Photodetectors Based on CVD-Grown Graphene and PbS Quantum Dots with Ultrahigh Responsivity[END_REF] but the concept was unsuccessful because the dark current was even larger than in a conventional film of CQDs. The concept was then revisited by replacing graphene by MoS2 to introduce a gap and reduce the dark current. Simultaneously, the absorption was pushed from telecom range to longer wavelengths by replacing the PbS CQDs by HgTe nanocrystals [START_REF] Huo | MoS2-HgTe Quantum Dot Hybrid Photodetectors beyond 2 µm[END_REF]. This strategy keeps suffering from two main limitations, which are the fast saturation of the optical response (responsivity is only large under zero photon flux) and large memory effects (ie long time response).

Focal plane array 4.1. Pixel downsizing and coupling to read out circuit

When it comes to building a focal plane array, the pixel size matters. Most of current devices reported in the literature are based on a chip with several individual devices, each of them being typically around 1 mm 2 area. When such size is far too small in the case of solar cells where m 2 are required, it is also far too big for use in cameras, where the pixel should be ideally just above the targeted wavelength to improve image quality. Current IR technologies present pixel sizes in the 10 to 50 µm range. These values are actually limited by technology and more specifically by the indium bump hybridization to the read-out circuit, which becomes very complex for size below 10 µm. There is, in this sense, a true opportunity for quantum dots here [START_REF] Malinowski | Thin-Film Quantum Dot Photodiode for Monolithic Infrared Image Sensors[END_REF]. Because the active layer can be directly deposited on a CMOS read-out circuit, demonstration of pixel sizes below 10 µm will bring a significant advantage to CQDs. This nevertheless has still to be demonstrated and also raises new questions such as the ability to design large scale homogeneous films while preventing pixel cross overs. In the hypothesis where optical coupling between pixels will prevent the use of a continuous film of CQD, the film will have to be etched to effectively split the pixels (Lhuillier et al., 2016b). In this case, the material will have to sustain a lithography step (ie high temperature exposure and exposition to solvent).

Stability and encapsulation

Long term stability of CQD based device is a problem which has been mostly swept under the rug. It is conveniently admitted that stability of CQDs is higher than the one of organic materials, however there is a clear lack of data regarding this question. Probably absorption is a much more robust property than photoluminescence, nonetheless lead chalcogenides get oxidized and quickly get a PbO shell, when mercury chalcogenide are also air sensitive (Jagtap et al., 2018a;[START_REF] Lhuillier | Mid-Infrared HgTe/As2S3 Field Effect Transistors and Photodetectors[END_REF] even if the exact mechanism remains unclear. Two paths can be followed to address this question: either tuning the surface chemistry to make the CQD stable in air, or processing the material in air-free conditions followed by the encapsulation of the CQD-based device below a protective layer. Handling of CQDs in glove box has become more or less the regular procedure in the field. Far less work has been devoted to the question of encapsulation. Certainly, concepts from the field of organic electronics and CQD based solar cells [START_REF] Tan | Stabilities Related to Near-Infrared Quantum Dot-Based Solar Cells: The Role of Surface Engineering[END_REF]) can be reused. This include deposition of encapsulation polymers such as CYTOP, ALD (atomic layer deposition) deposition of thick alumina layer [START_REF] Ihly | The Photothermal Stability of PbS Quantum Dot Solids[END_REF] or nanoparticle shelling [START_REF] Durmusoglu | Production of Small, Stable PbS/CdS Quantum Dots via Room Temperature Cation Exchange Followed by a Low Temperature Annealing Processes[END_REF]. Nevertheless, it is critical to consider that current PbX and HgX materials are synthetized at low temperature (<150 and <100°C, respectively) and that processing them at temperature higher than their growth temperature will undoubtedly lead to a significant sintering and its associated dark current rise. As a result, specific low temperature methods need to be developed.

Recently (Jagtap et al., 2018a) have reported the low temperature deposition of combination of water-proof (PMMA and PVDF) and oxygen-proof (PVA) layers leading to stability over at least three months. This is comparable to the stability obtained for solar cell based on PbS CQD [START_REF] Chuang | Improved performance and stability in quantum dot solar cells through band alignment engineering[END_REF]. Longer stability investigation (at least up to one year) under realistic operational environment will have to be done to confirm the potential of the technology.

CONCLUSION

Thanks to 10 years of intensive research, IR CQDs have undoubtedly reached a maturity level where it can be considered as a possible alternative to historical semiconductors for IR sensing. Main achievements include full tunability of the absorption over the IR range, BLIP photodiode, demonstration of CQD based focal plane array in both SWIR and MWIR range. We have tentatively listed the main technological challenges that still need to be addressed to fully transfer this technology to industry.

Figure 1

 1 Figure 1 a. Scheme of a photoconductive device in planar geometry. b. I-V curve of a photoconductive device under dark condition and under illumination. c Scheme of a photodiode in vertical geometry. TCO, ETL and HTL stands respectively for transparent conductive oxide, hole transport layer and electron transport layer. d. I-V curve of a photodiode under dark condition and under illumination.

Figure 2 :

 2 Figure2: Illustration of the main challenges that need to be addressed to bring the field of IR nanocrystals to a mature level for thermal imaging.

Figure 3

 3 Figure 3 a, b and c are respectively the scheme for interband, intraband and plasmonic transitions in nanocrystals. d. Absorption spectra for mercury chalcogenide (HgSe and HgTe) nanocrystals of various sizes

ACKNOWLEDEGMENTS

EL thanks the support ERC starting grant blackQD (grant n°756225). We acknowledge the use of clean-room facilities from the "Centrale de Proximité Paris-Centre". This work has been supported by the Region Ile-de-France in the framework of DIM Nano-K (grant dopQD). This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE and also by the grant IPER-nano2. NG and JR thank Nexdot for post doc funding. We thank N. Pere-Laperne, A. Nedelcu and J.-L. Reverchon for their valuable feedback on the manuscript.


Raymond A Marie
email: raymond.marie@irisa.fr

THE IMPACT OF THE UNAVAILABILITY OF LINE REPLACEABLE UNITS ON THE OPERATIONAL AVAILABILITY OF SYSTEM

Keywords: integrated logistic system, stock shortage, line replaceable unit, operational availability, intrinsic availability, Markovian model, approximate method, non product form queuing network

The evaluation of the operational availability of a fleet of systems on an operational site is far from trivial when the size of the state space of a faithful Markovian model makes this issue unrealistic for many large fleet of systems. The main difficulty comes from the existence on the site of line replaceable units that may be unavailable from time to time when a breakdown occurs. In this paper, we present a simpler but approximate method which has given a quite good accuracy when tested on small cases. The main idea is to consider a non product form queuing network and to aggregate subsets of it as if they were parts of a product form queuing network. Nevertheless, the generalization to systems with multiple types of line replaceable units needs to be investigated further and tested with respect to the accuracy of the new approximate method.

INTRODUCTION

Engineers who design new equipment do everything in their power to make this equipment reliable. This means they aim to achieve a low failure rate; possibly at the cost of very good quality components and / or by the establishment of redundant structures in the sense of reliability. The limits of this approach being, on the one hand the level of their expertise and on the other hand, the cost of the future equipment. In this study, we focus on the performance of (often complex) systems used in multiple copies on the same operational site and therefore on the prediction of the operational availability of the fleet assigned on the site. As far as possible during the design phase, the reliability engineers have estimated an intrinsic availability.

The intrinsic availability is an upper limit of the operational availability because the considered unavailability corresponds only to the time necessary to proceed with the exchange of the defective element. This assumes that the repairer and the spare part are always immediately available on the operational site. To reduce intervention time at an operational site, system repair consists of replacing the defective subset with an identical one in good condition. These exchangeable subsets are called LRUs (for "line replaceable units"). Restoring a piece of system by exchanging an LRU makes it possible to obtain a rapid return to service of the system and does not require the presence on the operational site of several specialists expert in their own field of expertise. Thus, we can change an aircraft engine in a few hours thanks to stakeholders who are knowledgeable about the procedures for intervention on fasteners and connections but who are not required to know the techniques to repair a broken engine. The operational availability (the one that is of interest to the user) will generally be lower than the intrinsic availability because the latter integrates the unavailability of repairer or LRU, if any. And if the waiting time for a repairer is generally measured in hours, the unavailability of a LRU can be measured in days, in weeks, even in months! It is therefore this last point which a potential customer should worry about in priority. Moreover, the unavailability of the LRU is, as we shall see, more difficult to model and evaluate than that of the repairer.

To try to predict the operational availability of a future fleet of systems, several directions of research can be considered. Thus, a simulation of the logistic support system, represented by a discrete event system, can be considered; but the number of event types to consider and the total number of events to simulate (to obtain reliable results) will generally be ground for abandoning this direction. A second possible direction is that of a Markovian model of the support system. But if this path is attractive to deal with the case of a subsystem, there is an explosion in the number of possible states when trying to deal with the case of many support systems. There remain the directions of the so-called approximate methods which can be very acceptable when the parameters remain confined to a given subspace of the physically admissible space of the said parameters. Unfortunately, the acceptable subspace seems very difficult to characterize! This is why in this study we consider a virtual system with only one type of LRU in order to understand the influence of various parameters on the operational availability both of a faithful Markovian model (for which we are able to get the exact answer) and of a proposed approximate method. By doing so, we are able to compute the relative errors induced when using this latter solution. Note that if the spare LRUs were always available on the operational site (which is never the case) we could fairly model the behavior of the support system on the site by means of a so called product form queueing network, a well known class of queueing networks.

The paper is organized as follow. In a first step (Section 2), we build a Markovian model to take into account the presence of a limited stock of LRUs on the site. The model is relatively simple since, as we said, we consider a unique type of LRU. Then we use this model to better understand the variation of the operational availability according to the different parameters (Section 3). In Section 4, we first introduce a new approximate method for calculating the operational availability and then we test the influence of the parameters on the relative error thanks to the Markov model. Finally, we conclude this paper by summerizing the obtained results and by mentionning potential extensions.

DETERMINATION OF A MARKOVIAN MODEL

As mentionned in the introduction, we only consider the case of a single type of LRU equipment in order to have a probabilistic model that is as simple as possible while behaving in a way that is almost similar to a real system. Note that the LRU equipment is the only subset of the virtual system that can break down. Suppose the fleet consists of M (very small) systems. Let n 1 , n 2 and n 3 be the number of systems that are, at a given time, 1) in operating state, 2) unavailable and waiting for a new LRU and 3) unavailable and in the process of exchanging their LRU, respectively. We assume that the operating time and the exchange time of an LRU follow exponential probability distributions of respective λ and µ rates. Let's define s as the number of repairers assigned to support this fleet. To reduce the waiting time for a LRU in good condition, a (small) stock of R LRUs in good condition is initially assigned to to the operational site.

The modeling and the evaluation of the expectation of a potential waiting time for a spare LRU is less straightforward. To introduce the difficulty, let us begin by presenting an acceptable Markovian model which would give good results if the exchange rate µ was infinite (this assumes that n 3 is always null). In such a situation, the Markovian model is a Birth and Death process (BDP) whose possible states are the integers 0, 1, 2, ..K, where K = M + R. Here, breakdowns of systems are associated to death events of the BDP and the order deliveries are associated to birth events of the BDP. Given that the BDP is in state k, then n 1 , the number of systems in operating state equals (M -(M -k) + ), where (M -k) + = max(0, (M -k)). The transition rate of a death event equals n 1 λ, i.e., (M -(M -k) + )λ. Consequently, n 2 , the number of systems wanting for a LRU in good condition, equals (M -n 1 ) = (M -k) + when this BDP is in state k. Also the number of LRUs awaiting delivery equals (K -k). Here we assume that the transition rate β(k) of a birth event from state k to state (k + 1) depends only on (K -k), the number of LRUs awaiting delivery.

The determination of transition rate β(k) of a birth event is not straightforward and is developed thereafter. First, let us imagine a process running through a source-terminal diagram describing the timed steps that are produced following the issuance of the order of an LRU, until delivery to the operational site of the LRU. This process has a random duration which corresponds to the travel duration on an oriented random graph. Now imagine that delivery of the LRU immediately triggers a new LRU order. We would then have a loop traversed by an order whose rate of passage on the operational site would be the opposite of the expected duration of travel of the loop. Let's define Beta(1) this rate of transition. Suppose now that k orders circulate in this loop permanently. The rate of passage on the operational site of these orders, noted Beta(k), will be greater than Beta [START_REF] Marie | Modèlisation par Réseaux de Files d'Attente[END_REF]. We assume that Beta(k + 1) ≥ Beta(k) whatever is k and that Beta(k) ≤ kBeta(1). The case Beta(k) = kBeta(1) corresponds to an organization whose shared resources are sufficient to avoid waiting time (for example the faulty LRU is not blocked due to the unavailability of a repairer).

Let's suppose that thanks to a rough model solved by pen and paper or by simulation, we can estimate the values of the Beta(k), k = 1, 2, ..., K; then the procedure is to take β(k) = Beta(K -k), k = 0, 1, 2, ..., (K -1). In this paper, in order to facilitate the interpretation of the results, we examine the specific situation where [START_REF] Marie | Modèlisation par Réseaux de Files d'Attente[END_REF]. This means that Beta(1) will be a parameter that we will denote β. The Markovian graph of this model is represented in Figure 1. Now that we have worked out the model relative to the LRU delivery mechanism, we consider the overall model which includes the variable n 3 (when µ is finite). This overall model is a continuous time Markov chain (CTMC) such that a state of this CTMC corresponds to a couple (n 1 , k) belonging to the set

β(k) = (K -k)Beta
0 1 2 k M K • • • • • • • • • λ β(K) 2λ β(K -1) β(K -2) β(K -k) β(K -M ) (M -(M -k) + )λ M λ M λ
{(n 1 , k) | 0 ≤ n 1 ≤ M, n 1 ≤ k ≤ K}. Note that if n 1 and k are known, then n 3 = M -n 1 -(M -k) +
. This is the number of unavailable systems for which the repairers are in the process of exchanging their LRU.

If a e;e ′ denotes an element of the infinitesimal generator of the CTMC, then the non-null non-diagonal elements are given by the following relations:

a (n 1 ,k);(n 1 -1,k) = n 1 λ if k > M (1)
a (n 1 ,k);(n 1 -1,k-1) = n 1 λ if k ≤ M a (n 1 ,k);(n 1 ,k-1) = β(k) if k < K a (n 1 ,k);(n 1 +1,k) = µ(n 3 ) if n 3 > 0 , where µ(n 3 ) = inf(s, n 3 )µ and n 3 = M -n 1 -(M -k) + .
The knowledge of this infinitesimal generator allows us to compute the steady-state probability distribution of the CTMC [START_REF] Sericola | Markov chains. Theory, Algorithms and Applications[END_REF][START_REF] Stewart | Introduction to the Numerical Solution of Markov Chains[END_REF][START_REF] Trivedi | Probability and Statistics with Reliability, Queuing, and Computer Science Applications[END_REF] and from that to determine the steady-state availability. This solution will serve as the reference point when we will be testing the approximate method that will be introduced in Section 4.

In Section 3 below, we use this first model to examine its sensitivity with respect to different parameters.

SENSITIVITY WITH RESPECT TO THE DIFFERENT PARAMETERS

If not mentioned otherwise, we will consider the normalized availability, i.e., the mean number of available systems divided by M , the total number of systems corresponding to the fleet.

We first look at the sensitivity of the Markovian model with respect to M when the total failure rate M λ stays constant.

In Figure 2 we investigate the influence of M on the availability when the product M λ stays constant and M increases from M = 1 to M = 50. The product M λ stays equal to 0.06. Parameters β and µ are respectively equal to 0.05 and 0.2. The availability is plotted for R = 0, 1, 2. As expected, the availability increases with the value of R. When M = 1 and R = 0, the mean operational time equals 1/λ = 1/0.06 while the mean non-operational time equals (1/β + 1/µ) = (1/0.05 + 1/0.2). The corresponding value of the availability is 0.4. When M = 50 and R = 0, the mean operational time equals 1/λ = 1/0.012 while the parameters β and µ keep the same values. Since the total failure rate does not increase, the unique repairer keeps a small utilization ratio and the mean rate of orders of LRU does not increase either. As a consequence, the mean non-operational time does not increase significantly while the mean operational time is multiplied by 50. This involves a significant increase of the availability.

In Figure 3 we investigate the influence of M on the percentage of unavailability due to the lack of good LRUs again when the product M λ stays constant and M increases from M = 1 to M = 50. We plotted the ratio of the unavailability due to the lack of good LRUs over the total unavailability. When M = 1 and R = 0, this ratio corresponds to (1/β)/(1/β + 1/µ) which is here equal to 0.8. We can see in this figure that the ratio decreases when M increases. This is due to the fact that when R = 0, the percentage corresponding to the delivery of good LRUs behaves as a queue with an infinite number of servers while the limited number of assigned repairers induces some extra waiting time in the exchange process. When R takes a non-null value, the behaviour of the delivery of good LRUs does not behave as a classical queue and the evolution of the ratio w.r.t. M has no obvious explanation. In Figure 4 we consider the variation of parameter β from 0.006 to 0.8. We plotted the availability for R = 0, 1, 2 and 3. In this figure, the parameters with fixed values are M = 30, λ = 0.002, µ = 0.02 and s = 4. We remark that when β is large enough, the availability tends to a maximal value of 0.893 that corresponds to the limit when β tends to infinity. In that situation, the remaining unavailability is the consequence of the time to exchange the LRU. Note that when R = 3, this maximal value is practically reached when β = 0.02. In other words, assuming β = 0.02 and R = 3, then the lack of LRUs has a negligible influence on tne unavailability w.r.t. that of the exchanging time.

If we had plotted the variation of availability w.r.t. parameter µ, we would have observed a higher limit of the availability for R = 3 and β ≥ 0.02. For example, the availability equals 0.98 when µ = 0.1 and 0.998 when µ = 1.

Figure 1 :

 1 Figure 1: Transition graph of the BDP.

Figure 2 :

 2 Figure 2: Availability for R = 0, 1, 2. Influence of M when the product M λ stays equal to 0.06.

Figure 3 :

 3 Figure 3: Unavailability ratio for R = 0, 1, 2. Influence of M when the product M λ stays equal to 0.06.

Availability

Figure 4: Availability for R = 0, 1, 2 and 3. Influence of β.

A NEW APPROXIMATE METHOD

The model

We consider a queuing network with three stations. Station 1 models the operational systems, with an individual failure rate λ. Station 3 models the exchanging operation of the failed LRU where each repairer is a server with rate µ. These two stations are of the class of product form stations. Station 2 is the one that makes the queuing network not a product form one (except when R = 0).

The idea is to aggregate the sub-network composed of the two product form stations in order to create a single station as if these two stations were part of a product form queuing network. It is shown in [START_REF] Marie | Modèlisation par Réseaux de Files d'Attente[END_REF] that this aggregation would not modify the initial solution of the complementary network if the queuing network was a product form one. Following [START_REF] Marie | Modèlisation par Réseaux de Files d'Attente[END_REF], the service rate ν a (i) of the aggregated station can be expressed as:

where, in the special case of stations 1 and 3, c a is an arbitrary constant and

In addition, note that A 3 (0) = 1 and G(0) = 1.

Once the sequence ν a (i) has been computed, the last phase of the proposed method consists in studying the special station 2 that we introduced in Section 2 with a new arrival rate of failed LRUs:

In the following section, we test the accuracy of this approximate method using the solution given in Section 2.

Experimental results

Since this approach gives the exact availability when R equals to zero, we only consider positive values of R while performing this experiment.

In Figure 5 we examine the evolution of the relative error introduced by the proposed method on the availability when the parameter β varies from 0.006 to 0.8. We consider the cases R = 1, 2 and 3. We observe that the three curves are concave and that they realize their maximal values for distinct values of β. The higher maximal value is close to 10 -3 and occurs when R = 3. That would allow us to estimate very accuratly the operational availability of a real fleet of systems. In addition, in the real world on an operational site, the values of R are generally either R = 0 or R = 1 and rarely greater than one. That moderates the fact that the maximal value seems to be increasing w.r.t. parameter R. In this figure, the parameters with fixed values are M = 30, λ = 0.002, µ = 0.02 and s = 4. In Figure 6 we examine the evolution of the relative error introduced by the proposed method on the availability when the parameter µ also varies from 0.006 to 0.8. Let's consider once more the cases where R = 1, 2 and 3. We observe that the three curves are no longer concave but still uni-modal and that they realize their maximal values for values of µ which are very close to each other. The higher maximal value is also obtained when R = 3 and is close to 10 -3 . In this figure, the parameters with fixed values are M = 30, λ = 0.002, β = 0.02 and s = 4.

CONCLUSION

Due to the explosion of the state-spaces of classical Markovian models when evaluating operational availability of fleet of systems of real size, with a risk of shortage of LRUs, we presented a new approximate method. So far, this approach has been tested on a simple system reduced to a single type of LRU. We used a classical Markovian model (a continuous time Markov chain) to serve as a reference point to test the new approximate method. Furthermore the Markovian model furnished numerical results that contributed to understand the behavior of the logistic of the fleet of systems. This new approach gives promising results by providing low relative errors on the approximated operational availability. Hence the generalization to systems with multiple types of LRU needs to be investigated further and tested with respect to the accuracy of the new approximate method.