Chaaben Kouki
email: chaaben.kouki@gmail.fr

Oualid Jouini
email: oualid.jouini@ecp.fr

On the Effect of Lifetime Variability on the Performance of Inventory Systems

Keywords: Periodic review, Perishable items, Markov Process, Erlang lifetimes, inventory models

We consider a periodic review perishable inventory system. Item lifetimes are random and assumed to follow an Erlang distribution. This is a rich modeling allowing to cover various cases of lifetime variability in practice. The demand process is assumed to be Poisson and the order lead time is constant.

For the extreme cases of exponential and deterministic lifetimes, we provide an analytical approach for the system performance evaluation. We also conduct a simulation study to examine the important effect of lifetime variability mixed with the cost parameters on the total operating cost. It appeared that there might be opportunities to considerably improve the operating cost in the case of high lifetime variability. This is more apparent for cases with large outdate, lost sale, and purchasing cost parameters.

Introduction

The operations management of perishable inventory systems has been mainly stimulated by the economic impact of perishability. In the grocery and pharmaceutical industry, expiration is responsible of 19% and 20% of total unsaleable respectively (Joint Industry Unsaleables Benchmark Survey, 2008). [START_REF] Lystad | Single stage heuristic for perishable inventory control in two-echelon supply chains[END_REF] reported that about $30 billion are lost due to perishability in US grocery industry. Although the tradeoff between customer satisfaction and cost minimization could be handled through appropriate standard inventory control rules, the randomness of the item lifetimes makes such rules unsuccessful [START_REF] Karaesmen | Managing perishable and aging inventories: Review and future research directions. Planning Production and Inventories in the Extended Enterprise: A State of the Art Handbook[END_REF]. In this paper, we study the effect of the lifetime variability on the performance of inventory systems with perishable items. The application we have in mind is that of grocery products with no printed expiration date. Examples of such products include fresh fruits, vegetables, flowers and seafoods [START_REF] Xiao | Supply chain management of fresh products with producer transportation[END_REF]. The change in the storage conditions (temperature, humidity, light exposure, etc.) considerably affect an item lifetime. This may create in turn a high variability in the items remaining lifetimes. We focus here on the study of the impact of such variability on the system performance, i.e., to what extent such variability may affect the inventory system operating costs? The answer to this question would help managers to assess the usefulness of investing on improving storage conditions in order to reduce the lifetime variability.

In this paper, we address the problem of perishable inventory management where the lifetime is modeled as a random variable. This allows to capture real-life situations of lifetime time variability. We consider the periodic review policy (T, r, Q). The inventory level is periodically observed at the beginning of equal intervals of time with length T each. If, and only if, at the observation epoch the inventory level is at or below the reorder point r, a replenishment order of Q units is placed. As it is very often the case in practice, we choose a periodic review inventory scheme instead of a continuous one. For instance, hospitals typically order blood products from a central blood center periodically with regular order at the beginning of the ordering cycle and emergency order during the cycle [START_REF] Zhou | Inventory management of platelets in hospitals: Optimal inventory policy for perishable products with regular and optional expedited replenishments[END_REF].

Moreover, Material Requirement Planning and Automated Store Ordering System order a regular batch from suppliers at fixed intervals of time [START_REF] Shang | Optimal and heuristic echelon (r, nQ,T) policies in serial inventory systems with fixed costs[END_REF][START_REF] Van Donselaar | Ordering behavior in retail stores and implications for automated replenishment[END_REF]. Finally the (T, r, Q) policy allows a cost reduction of the ordering and transportation costs by ordering items together with a common base period of review.

We consider Erlang distributions for lifetimes, which allows to cover a wide range of variability by varying the number of phases. The coefficient of variation may vary from the extreme value 1 to the 1 extreme value 0. The first extreme value corresponds to the case of an exponential distribution (high variability). This is reasonable for systems where item lifetimes are typically small but there are occasionally long lifetimes. The second extreme value corresponds to the case of a deterministic distribution (no variability), namely cases with perfectly respected storage and transportation conditions. The main contributions of this paper can be summarized as follows.

• For the extreme cases of exponential and deterministic lifetime distribution, we develop an analytical approach for the system performance evaluation during the steady-state regime. Our approach for the exponential case relies on the use of embedded Markov chains, and that for the deterministic case relies on the use of the regenerative property of the underlying process. For the two extreme cases, we compute the expected inventory level, the rates of orders, perished items and lost sales.

Under an economic framework, this allows us next to optimize the total operating cost. To the best of our knowledge, the periodic review (T, r, Q) policy for perishable inventory systems with stochastic lifetimes has not been studied yet. In Table 1, we provide a comparison between the key papers dealing with perishable inventory systems. The comparison is based on the features of the considered models. In this paper, we consider a periodic (T, r, Q) replenishment policy, Poisson arrivals, lost sales, a deterministic lead time, a general lifetime distribution and an infinite planning horizon.

• For the general case with Erlang distributed lifetimes, we conduct a comprehensive simulation study that examines the important effect of the lifetime variability on the operating costs. It appeared that there might be opportunities for managers to considerably improve the operating cost in the case of high lifetime variability. This is more apparent for cases with large outdate, lost sale, and purchasing cost parameters.

The remainder of the paper is organized as follows. In Section 2, we review the literature of perishable inventory management related to this work. In Section 3, we describe the considered inventory model and concretely define the objective of the paper. In Section 4, we develop an analytical analysis for the particular cases of exponential and deterministic lifetimes. We also provide some numerical experiments to compare between them. In Section 5, we conduct a comprehensive analysis on the impact of lifetime variability using simulation. In Section 6, we give concluding remarks and highlight some future research. [START_REF] Kalpakam | Continuous review (s, S) inventory system with random lifetimes and positive leadtimes[END_REF] (s,

S)

Infinite

Continuous

Literature Review

The analysis of inventory systems with perishable items has received significant attention to date in the literature. The effect of perishability on inventory management is studied either by combining inventory management with discounted pricing policies, either by an investment on technologies that help to preserve items on stock, or by seeking for the best inventory replenishment policies. Among the studies focusing on dynamic pricing for perishables inventory systems, one may find [START_REF] Herbon | Perishable inventory management with dynamic pricing using timetemperature indicators linked to automatic detecting devices[END_REF] and [START_REF] Chew | Optimal dynamic pricing and ordering decisions for perishable products[END_REF], and [START_REF] Jia | Dynamic ordering and pricing for a perishable goods supply chain[END_REF]. Works dealing with investment on technologies to track/evaluate the item lifetime include those by [START_REF] Piramuthu | RFID and perishable inventory management with shelf-space and freshness dependent demand[END_REF], [START_REF] Grunow | RFID in highly perishable food supply chains remaining shelf life to supplant expiry date[END_REF], [START_REF] Wang | A dynamic product quality evaluation based pricing model for perishable food supply chains[END_REF][START_REF] Wang | A dynamic product quality evaluation based pricing model for perishable food supply chains[END_REF][START_REF] Dye | The effect of preservation technology investment on a non-instantaneous deteriorating inventory model[END_REF]. For investigations focusing on the best ordering policies, we refer the reader to [START_REF] Bakker | Review of inventory systems with deterioration since 2001[END_REF], [START_REF] Karaesmen | Managing perishable and aging inventories: Review and future research directions. Planning Production and Inventories in the Extended Enterprise: A State of the Art Handbook[END_REF][START_REF] Nahmias | Perishable Inventory Systems[END_REF] for comprehensive surveys on periodic and continuous review policies for perishable inventory systems. When focusing on perishable inventory systems under stationary demand distribution, one can classify them into various classes depending on whether the product lifetime is deterministic or stochastic, the review policy is periodic or continuous, and the replenishment lead time is positive or instantaneous.

For deterministic lifetime models, most of works consider a periodic review and use the dynamic programming approach to find the optimal ordering quantity. Such analyses highlight the difficulty to track the different age's categories of items in stock since more variables are needed to track these remaining items and their ages to find the optimal order quantity Nahmias (2011). To overcome this complexity, approximate policies have been proposed in literature. For instance, [START_REF] Haijema | Blood platelet production: Optimization by dynamic programming and simulation[END_REF][START_REF] Haijema | Blood platelet production with breaks: optimization by SDP and simulation[END_REF] and [START_REF] Haijema | A new class of stock-level dependent ordering policies for perishables with a short maximum shelf life[END_REF] use a combination of dynamic programming and simulation to reduce the state space and provide "near optimal" order-up-to-level inventory policies. The order-up-to-level policies are bounded by a maximum and a minimum and work only under the case of Poisson demand distribution. van Donselaar and [START_REF] Van Donselaar | Approximations for the relative outdating of perishable products by combining stochastic modeling, simulation and regression modeling[END_REF] focus on the periodic review (r, nQ) policy and propose an approximation of the outdating quantity by combining stochastic modeling, simulation and regression modeling.

Under a continuous review (s, S) policy and assuming instantaneous replenishment lead times, Weiss (1980) followed by [START_REF] Lian | Continuous review perishable inventory systems: Models and heuristics[END_REF] and [START_REF] Lian | A perishable inventory model with markovian renewal demands[END_REF] use the Markov renewal processes and the concept of regenerative processes in order to derive the optimal parameters. Heuristics to manage the case of positive lead times are proposed in [START_REF] Lian | Continuous review perishable inventory systems: Models and heuristics[END_REF], Chiu (1995b), [START_REF] Kouki | Assessing the impact of perishability and the use of time temperature technologies on inventory management[END_REF] and in van Donselaar and [START_REF] Van Donselaar | Approximations for the relative outdating of perishable products by combining stochastic modeling, simulation and regression modeling[END_REF]. [START_REF] Tekin | Age-based vs. stock level control policies for perishable inventory system[END_REF] introduce the age-based inventory policy which is suitable for particular items that start perishing when the order is unpacked for use. [START_REF] Duan | A new age-based replenishment policy for supply chain inventory optimization of highly perishable products[END_REF] proposed also an age-based policy to reduce waste of perishables. More recently, [START_REF] Berk | Analysis of the (Q,r) inventory model for perishables with positive lead times and lost sales[END_REF] derive a closed form expression of the total ordering cost under the (r, Q) inventory policy with lost sales and positive lead times. One of the analytical findings in this paper considers the case of deterministic lifetimes under the (T, r, Q) policy with positive and constant lead times, for which no results exist yet.

Most of the work, in the literature dealing with random lifetimes, considers exponentially distributed lifetimes. [START_REF] Kalpakam | Continuous review (s, S) inventory system with random lifetimes and positive leadtimes[END_REF] are the first to study an (s, S) model for a Poisson demand distribution where product lifetimes and lead times are assumed to follow exponential distributions. Under these assumptions, the underlying inventory process is Markovian. By assuming lost sales and restricting the number of outstanding replenishment orders to, at most, one at any given time, they derive the steady-state probabilities and obtained the exact cost function and some useful analytical properties regarding the reorder point. Similar models are also investigated in [START_REF] Kalpakam | Perishable system with modified base stock policy and random supply quantity[END_REF], [START_REF] Kalpakam | A perishable inventory system with modified (S-1, S) policy and arbitrary processing times[END_REF], [START_REF] Kalpakam | A continuous review perishable system with renewal demand[END_REF], [START_REF] Lian | A perishable inventory model with markovian renewal demands[END_REF] and in [START_REF] Kouki | Analysis of a periodic review inventory control system with perishables having random lifetime[END_REF].

Another new analytical finding in this paper considers the case of exponential lifetimes under the (T, r, Q) policy with positive and constant lead times.

There is an analogy between perishable inventory systems and queueing models with impatient customers. Known results from the queueing literature may be then used under certain conditions. For instance, [START_REF] Ioannidis | Control policies for single-stage production systems with perishable inventory and customer impatience[END_REF] studied a make-to-stock production systems that operate under an (S-1, S) policy for perishable item and impatient customers. The use of queuing models is initiated by [START_REF] Graves | The application of queueing theory to continuous perishable inventory systems[END_REF] who analyzes perishable inventory systems where customer and orders arrive according to a Poisson processes. When all demand requests are for the same quantity and without considering any explicit ordering policy, Graves shows that the inventory process is equivalent to the virtual waiting time process for an M/D/1 queue with customer abandonment. The similarity is easy to understand because the resupply time process can be seen as the server, the inventory as the queue, and the demand request as the customer arriving at the queue. If customers arrive according to a Poisson process and request an exponential batch size, the inventory process is equivalent to the virtual waiting time process for an M/M/1 queue with customer abandonment. [START_REF] Finch | Deterministic customer impatience in the queueing system GI/M/1[END_REF] shows that the sequence of waiting time of customers in the queue form a Markov chain with a finite state space. He derives the steady-state probability of the waiting time for any customer in the queuing system G/M/1+D. Kaspi andPerry (1983, 1984) introduce the concept of virtual death-process based on the analysis of an M/G/1 queue with impatient customer.

The virtual death-process is simply a reformulation of the age of the oldest item in stock used by [START_REF] Graves | The application of queueing theory to continuous perishable inventory systems[END_REF]. This concept is used by [START_REF] Perry | Perishable inventory systems with impatient demands[END_REF] who derive explicit expressions of the stationary distribution of two models where arrival of items and demands are state-dependent and customers are willing to wait. This work is generalized in [START_REF] Nahmias | Perishable inventory systems with variable input and demand rates[END_REF] by deriving the steady-state distribution of the virtual outdating process in the context where the demand rate depends on the current value of the basic virtual outdating process.

Modeling and Objective

Consider a stochastic inventory system with a single supplier, which makes a single product (item) type to stock in order to satisfy a random demand. The arrival of customers follows a Poisson process with rate λ. Each arriving customer requests one unit of product. The replenishment (lead) time is assumed to be deterministic and is denoted by L.

Products cannot be infinitely stored. Each stored item has a random duration of time before it perishes. The time to perish is referred to as the lifetime. A perished item in the stock is unusable and is then discarded from the system as soon as it perishes. Product lifetimes are independent, identically distributed and follow an m-Erlang distribution with rate δ, where m is the number of phases, m ≥ 1.

Under the m-Erlang distribution, the time to perish of an item consists of m phases, each phase is exponentially distributed with rate mδ. The expected time to perish for an item is then 1/δ. The family of m-Erlang distributions provides a modeling flexibility allowing therefore to cover various cases in practice. The exponential distribution is a special case of the m-Erlang distribution when m = 1. The exponential distribution is common in modeling randomness in inventory operations. It is reasonable for systems with high lifetime variability where lifetimes are typically small but there are occasionally long lifetimes. As m increases, the m-Erlang becomes more symmetrical, and as m approaches ∞, lifetime becomes deterministic. The m-Erlang distribution also allows us to model lifetimes with a wide range of variability. In particular, by increasing m while keeping δ constant, we can reduce the coefficient of variation of lifetimes (equal to 1/ √ m) while keeping the mean lifetime (equal to 1/δ) constant.

Upon arrival, a customer is immediately satisfied by one available stored item, if any. If not, the customer is lost, i.e., leaves the system without service (loss of goodwill). The stored items are sold under the first in, first out (FIFO) discipline of service. This means that items made to stock earlier are sold first. The motivation of the FIFO rule assumption comes from a common practice in supermarkets, as mentioned in [START_REF] Tekin | Age-based vs. stock level control policies for perishable inventory system[END_REF]. Often, only one age category is exposed on the shelves to customers, and the remaining inventory is kept in a separate storage area. Recall also that we consider in this

paper item examples with no printed expiration dates. For such items (vegetables, seafoods, etc.), it is often not easy for customers to identify the age category or the level of freshness. Therefore, even if items from different orders are mixed on the shelves, it suffices for the supermarket to put the items on the shelves such that the more recently arrived items are the less accessible ones to customers. This makes FIFO an appropriate assumption. Finally note that if there is complete information about residual lifetimes, the FIFO rule may not be an optimal policy (the shortest remaining lifetime first should behave better). However such information on residual lifetimes, of the items carried in stock, would require the use of advanced technology allowing to continuously update the remaining lifetime as a function of the evolution of some environment parameters [START_REF] Piramuthu | RFID and perishable inventory management with shelf-space and freshness dependent demand[END_REF][START_REF] Grunow | RFID in highly perishable food supply chains remaining shelf life to supplant expiry date[END_REF].

Such a technology is often expensive and then not available in most cases in practice. With no updated information on remaining lifetimes, FIFO is an appropriate policy. It enables to remove in expectation the oldest stored items, which allows to keep in stock the items with longer expected remaining lifetimes.

The system employs a periodic (T, r, Q) policy to control the inventory level. It works as follows. The inventory level is periodically observed at the beginning of equal intervals of time with length T each. If, and only if, at the observation epoch the inventory level is at or below the reorder point r, a replenishment order of Q units is placed. The inventory level is then ranging from 0 to r + Q. For tractability issues, we assume in this paper that the period length T is such that L ≤ T . This ensures that, at most, one order is outstanding at any point of time. In using this assumption, we are following longstand practice in the modeling of perishable inventory models such as in [START_REF] Tekin | Age-based vs. stock level control policies for perishable inventory system[END_REF] and in [START_REF] Berk | Analysis of the (Q,r) inventory model for perishables with positive lead times and lost sales[END_REF]. Without loss of generality, we assume that the inventory level is initially Q. An illustration of the inventory under the (T, r, Q) policy is provided in Figure 1.

We build an economic framework as follows. The system incurs a fixed ordering cost K per order, a purchasing cost C per item unit, a holding cost H per time unit for each stored item, an outdate cost W per each perished item, and a lost sale cost B for each customer arriving at an empty stock (this cost parameter reflects a loss of goodwill).

We next formulate a cost minimization problem. For 0 ≤ i ≤ r + Q, let us denote by P (i) the long run probability that the inventory level is i just after an eventual order arrival, i.e., at the time points {nT + L, n = 0, 1, 2...}. We denote by p i,j (t) the transition probability from state (inventory level) i to state j during an interval of time with length t, for 0 ≤ i, j ≤ r + Q. Averaging over all possibilities, the probability that an order is trigged at an arbitrary review point of time (beginning of an arbitrary period with length T ) may be then written as

r+Q ∑ i=0 r ∑ j=0 P (i) • p i,j (T -L). We also define E(I), E(O) and E(S)
as the long run expected inventory level, the long run expected rate of outdating items and the long run expected rate of lost sales, respectively. Hence, the total long run cost rate of the system, denoted by

T C(T, r, Q), is given by T C(T, r, Q) = K + C • Q T r+Q ∑ i=0 r ∑ j=0 P (i) • p i,j (T -L) + H • E(I) + W • E(O) + B • E(S). ( 1
)
The objective is to find the optimal values of the policy parameters T , r and Q, which minimize the steady-state total cost rate T C(T, r, Q). We also want to examine the effect of the variability of the lifetime distribution on this optimal cost. For ease of referencing, Table 2 provides a summary of the notations used throughout the paper.

Analytical Analysis of Particular Cases

The analytical analysis of the model described in Section 3 is too complex. We consider in this section two particular cases of exponential (high variability) and deterministic (zero variability) lifetime distributions, for which, we develop exact results. The objective of this analysis is to obtain some sense on the effect of the lifetime variability on the total optimal cost. This allows also to obtain lower and upper bounds of the total optimal cost for the general case. A more comprehensive analysis is then conducted in Section 5 using simulation. Expected inventory level per unit of time T * , r * , Q * Optimal review period, optimal reorder level and optimal order quantity, respectively T C * Optimal total cost rate of the system pi,j(t) :

Transition probability from state (inventory level) i to state j during an interval of time with length t P (i)

Steady state probability that the inventory level is i just after an eventual order arrival

The Case of Exponential Lifetimes

We consider the case of exponentially distributed lifetimes and develop an exact method to compute the steady-state total cost rate of the system. The method is based on the analysis of an embedded continuous-time Markov chain seen on the epoch when an order may arrive, from which we derive the steady-state probabilities of the inventory level. This allows to derive all the cost components thereafter.

Let us define the stochastic process, denoted by I(t), of the on-hand inventory level for the specific intervals of time nT + L ≤ t < (n + 1)T + L. Recall that a replenishment may only occur at the epochs {nT + L, n = 0, 1, 2...}. During an interval nT + L ≤ t < (n + 1)T + L, the events that may occur are either customer arrivals or product deteriorations. Since inter-arrival times and lifetimes are exponential, I(t) is a continuous-time Markov chain. More specifically, it is a generalized pure death-process. Let us define the transition probability p i,j (t

) = P {I(t) = j | I(nT + L) = i, nT + L ≤ t < (n + 1)T + L}, for 0 ≤ i, j ≤ r + Q and n ≥ 0.
Writing the Kolmogorov's Forward Equations implies

dp i,j (t) dt =            -(λ + iδ)p i,j (t) if i = j, -(λ + jδ)p i,j (t) + (λ + (j + 1)δ)p i,j+1 (t) if 0 ≤ j ≤ i -1, 0 otherwise, (2)
for 0 ≤ i, j ≤ r + Q and nT + L ≤ t < (n + 1)T + L. We denote by p i,j (z), the Laplace transform of p i,j (t), for z ∈ 4 + . Applying the Laplace transform to Equations (2) leads to

     (z + λ + iδ)p i,j (z) = 0 if i = j, (z + λ + jδ)p i,j (z) = (λ + (j + 1)δ)p i,j+1 (z) if 0 ≤ j ≤ i -1, (3)
for 0 ≤ i, j ≤ r + Q and z ∈ 4 + . Taking the initial condition p i,i (0) = 1, for 0 ≤ i ≤ r + Q, and recursively solving Equations (3), we obtain

p i,j (z) = 1 λ + jδ i ∏ k=j λ + kδ z + λ + kδ , ( 4
)
for 0 ≤ i, j ≤ r + Q and z ∈ 4 + . This equation has i -j simple poles at -(λ + kδ), k = j, j + 1, ..., i.

The residue at these poles can be expressed as

(z + λ + kδ) p i,j (z)| z=-(λ+kδ) = e -(λ+kδ)t k-1 ∏ n=j 1 (n -k)δ i ∏ n=k+1 1 (n -k)δ . ( 5
)
Using Equation ( 5) and after some algebra, the inversion of Equation ( 4) implies

p i,j (t) =                        e -(λ+iδ)t (e δt -1) i-j i ∏ k=j+1 (λ + kδ) (i -j)! δ i-j if 0 < j ≤ i -1, 1 - i ∑ k=1 p i,k (t) if j = 0, 0 otherwise, (6)
for 0 ≤ i, j ≤ r + Q and nT + L ≤ t < (n + 1)T + L. This characterizes the transition probability from state i at time nT + L to state j at time t < (n + 1)T + L. Let us now compute the transition probability p i,j (T ) from state i at time nT + L to state j at time (n + 1)T + L. The difficulty here is that a replenishment may occur at time (n + 1)T + L. We define a i,j (L) as the transition probability from state i at time (n + 1)T to state j at time (n + 1)T + L just after a replenishment (if an order is triggered at time (n + 1)T ). Starting from state i at time nT + L, the probability transition from i to j at time (n + 1)T is given by p i,j (T -L). Now, at time (n + 1)T + L the replenishment of size Q occurs if the state of the inventory position is less or equal to r at time (n + 1)T . Using Equation ( 6), a i,j (L) may be written as

a i,j (L) =            p i,j-Q (L) if 0 ≤ i ≤ r, Q ≤ j ≤ r + Q, j -i ≤ Q, p i,j (L) if r + 1 ≤ i ≤ r + Q, 0 ≤ j ≤ i, 0 otherwise. (7)
We are now ready to compute p i,j (T ). Averaging over all possibilities, it is given by

p i,j (T ) = i ∑ k=0 p i,k (T -L)a k,j (L), (8)
for 0 ≤ i, j ≤ r + Q.

Steady-State

U =          1 p 1,0 (T -L) p 1,1 (T -L) 0 . . . . . . . . . p r+Q,0 (T -L) • • • • • • p r+Q,r+Q (T -L)          , V =    0 V 1 V 2    , (9)
where V 1 and V 2 are the transition matrixes of size (r + 1) × (r + 1) and Q × (r + Q + 1), respectively.

They can be written as

V 1 =          1 p 1,Q (L) p 1,Q+1 (L) 0 . . . . . . . . . p r,Q (L) • • • • • • p r,r+Q (L)          (10) V 2 =          a r+1,0 (L) • • • a r+1,r+1 (L) a r+2,0 (L) • • • • • • a r+2,r+2 (L) 0 . . . . . . . . . . . . a r+Q,0 (L) • • • • • • • • • a r+Q,r+Q (L)          (11)
Thanks to the finite space, the numerical solving of the equation P = P • U • V is straightforward. This finishes the characterization of P (i) for 0 ≤ i ≤ r + Q.

Expected Operating Costs. Consider the time interval with length T that starts at time nT + L (just after a replenishment, if any) and ends at (n + 1)T + L (just before a replenishment, if any), for n ≥ 0. The inventory level at time nT + L is i with probability P (i), for 0 ≤ i ≤ r + Q. At any time t, nT + L ≤ t < (n + 1)T + L, it drops down to the level j, 0 ≤ j ≤ i, with probability p i,j (t), because of customer arrivals or product deteriorations. The conditional inventory level at that time t is then j,

given that the inventory level is i at time nT + L. Averaging over all possibilities for state i, the expected inventory level per unit of time is thus given by

E(I) = 1 T r+Q ∑ i=0 P (i)   i ∑ j=0 j ∫ T 0 p i,j (t)dt   . ( 12
)
Using the Markovian property of lifetimes, it follows that the expected rate of perished items is simply the expected inventory level times the lifetime rate. Hence

E(O) = δE(I). (13)

Next, we compute the expected rate of lost sales. Following [START_REF] Hadley | Analysis of Inventory Systems[END_REF], we first compute the length of time per cycle T (between time points nT + L and (n + 1)T + L) during which the system is out of stock. We denote by T S this time of stockout, 0 ≤ T S ≤ T . The system starts with an inventory level i at the beginning of the cycle, i ≥ 0. We distinguish two cases. The first case is for i = 0, for which we obviously have a stockout duration equal to the cycle duration, i.e., T . The second case is for i ≥ 1.

In this case, if the system reaches an out of stock condition in the time interval t to t + dt, it means that in the time 0 to t, i -1 items have been either perished or asked by customers and the i th item is perished or asked by a customer between t and t + dt. This occurs with probability p i,1 (t) • (λ + δ)dt. If the system does reach an out of stock position between t and t + dt, it will be out of stock for a duration T -t during the cycle T . The above arguments lead to

T S = P (0) • T + r+Q ∑ i=1 P (i) (∫ T 0 (λ + δ) • (T -t) • p i,1 (t)dt ) . ( 14
)
The expected number of lost sales during T is then the expected arrival rate times T S . Therefore

E(S) = λT S T , ( 15
)
which implies

E(S) = λP (0) + λ T r+Q ∑ i=1 P (i) (∫ T 0 (λ + δ) • (T -t) • p i,1 (t)dt ) . ( 16
)
Coming back to Equation (1), we obtain

T C(T, r, Q) = K + CQ T r+Q ∑ i=0 r ∑ j=0 P (i) • p i,j (T -L) + H + δW T r+Q ∑ i=0 i ∑ j=0 ∫ T 0 P (i) • j • p i,j (t)dt + λ • B • P (0) + λB T • r+Q ∑ i=1 ∫ T 0 P (i) • (λ + δ) • (T -t) • p i,1 (t)dt. ( 17
)
Note that due to the memoryless property of lifetimes, the outdate cost W in Equation ( 17) appears only in the element H + δW . For a given lifetime rate, this means that W may be simply defined as a part of H. This completes the performance characterization under the case of exponentially distributed lifetimes.

The Case of Deterministic Lifetimes

We focus here on the case of deterministic lifetimes, i.e., we assume that an item perishes after exactly 1 δ units of time. We distinguish three different cases: 1)

1 δ + L ≤ T , 2) 1 δ ≤ T < 1 δ + L and 3) T < 1 δ .
For the first two conditions, we provide an exact method to compute the performance measures using the regenerative property of the inventory level at the replenishment epochs. The last case requires to keep track of the remaining lifetimes of items from different previous orders, which makes the analysis too complex.

Investigations focusing on how to keep track of the inventory on-hand for each lifetime level are described in [START_REF] Karaesmen | Managing perishable and aging inventories: Review and future research directions. Planning Production and Inventories in the Extended Enterprise: A State of the Art Handbook[END_REF] and in [START_REF] Nahmias | Perishable Inventory Systems[END_REF]. For instance, [START_REF] Lian | Continuous review perishable inventory systems: Models and heuristics[END_REF] and [START_REF] Schmidt | S-1, S) policies for perishable inventory[END_REF] have demonstrated that a rigorous analysis would not be possible due to the excessive high dimensionality of the problem. This explains the emergence of several papers that take simplified assumptions such as a zero lead time, or approximations by for example ignoring the age of the "old" inventory at decision epochs. Such references include Chiu (1995a,b) who ignore the age distribution of the stored items. They only consider the information of the total inventory level at a decision epoch to derive approximate policies. [START_REF] Lian | Continuous review perishable inventory systems: Models and heuristics[END_REF] assume a zero lead time which allows to characterize their (s, S) policy parameters. They next suggested a heuristic to deals with positive lead time. Further references for similar treatments can be found in [START_REF] Liu | s, S) continuous review models for products with fixed lifetimes[END_REF], [START_REF] Gürler | Analysis of the (s, S) policy for perishables with a random shelf life[END_REF], van Donselaar and Broekmeulen (2012), and [START_REF] Haijema | A new class of stock-level dependent ordering policies for perishables with a short maximum shelf life[END_REF]. In order to keep an exact analysis,

we analytically analyze in this section the cases where the order Q is totally exhausted before the arrival of a new order, i.e., for T ≥ 1 δ . This ensures that a single age category of items at decision epochs. To overcome the difficulty of the analytical analysis of the case T < 1 δ , we resort to simulation in Section 5.

Following the renewal reward theorem in [START_REF] Ross | Applied Probability Models with Optimality Applications[END_REF], the expected rate of the total operating costs could be formulated as the expected cycle cost over the expected cycle length. In what follows, we compute this quantity for the two cases

1 δ + L ≤ T and 1 δ ≤ T < 1 δ + L.
The case 1 δ + L ≤ T . In this case, it is easy to see that the inventory level at the order arrival epoch will always start with a fresh batch Q. The regenerative cycle is therefore equal to T . After its arrival to the stock and prior to the time duration 1 δ , a removed item is necessarily asked by a customer (it has not reached its lifetime threshold). Recall that the arrival process is Poisson with rate λ. We then have

E(I) = Q ∑ j=0 j ∫ 1 δ 0 p Q,j (t)dt = Q ∑ j=0 j ∫ 1 δ 0 (λt) Q-j e -λt (Q -j)! dt = Q ∑ j=0 (Q -j) ∫ 1 δ 0 (λt) j e -λt j! dt. ( 18
)
Knowing that λ

∫ t 0 (λt) j e -λt j! dt = 1 - j ∑ i=0 (λt) i e -λt i!
, for any positive real t and any integer j, Equation ( 18) becomes

E(I) = Q ∑ j=0 (Q -j) 1 λ ( 1 - j ∑ i=0 (λ/δ) i e -λ/δ i! ) (19) = Q(Q + 1) 2λ - 1 λ Q ∑ j=0 j ∑ i=0 (Q -j) (λ/δ) i e -λ/δ i! .
Similarly, we can write

E(O) = Q ∑ j=0 (Q -j) • p Q,j ( 1 δ ) = Q ∑ j=0 (Q -j) (λ/δ) Q-j e -λ/δ (Q -j)! = Q ∑ j=0 (λ/δ) j e -λ/δ (j -1)! . ( 20
)
Also

E(S) = λ(T - 1 δ ) + λ ( ∫ 1 δ 0 λ( 1 δ -t) • p Q,1 (t)dt ) = λ(T - 1 δ ) + λ ( ∫ 1 δ 0 λ( 1 δ -t) e -λt (λt) Q-1 (Q -1)! dt ) , ( 21
)
or equivalently after some algebra,

E(S) = λT -Q -( λ δ -Q) Q-1 ∑ j=0 (λ/δ) j e -λ/δ j! + (λ/δ) Q e -λ/δ (Q -1)! . ( 22
)
Finally, the total operating cost is given by

T C(T, r, Q) = K + C • Q + H • E(I) + W • E(O) + B • E(S) T . ( 23
)
The Case

1 δ ≤ T < 1 δ + L.
The main difference with the previous case is that here the regenerative cycle length depends on whether an order is triggered or not T -L units of time after an order arrival.

Then, the cycle length is T with the probability that at least Q -r items are removed from the stock, i.e., with probability 1 -

Q-r-1 ∑ j=0 (λ(T -L)) j e -λ(T -L) j!
, and it is 2T with the complementary probability.

Let E(Cycle Length) denotes the expected length of the regenerative cycle. It is then given by

E(Cycle Length) = T   1 - Q-r-1 ∑ j=0 (λ(T -L)) j e -λ(T -L) j!   + 2T   Q-r-1 ∑ j=0 (λ(T -L)) j e -λ(T -L) j!   = T   1 + Q-r-1 ∑ j=0 (λ(T -L)) j e -λ(T -L) j!   . ( 24
)
Similarly to the previous case, the expected inventory level per time unit and the expected rate of perished items are given by Equations ( 19) and ( 20), respectively. The expected rate of lost sales is however different because it depends on the cycle length. By conditioning on the cycle length duration, it may be written as

E(S) = λ   1 - Q-r-1 ∑ j=0 (λ(T -L)) j e -λ(T -L) j!   ( T - 1 δ + ∫ 1 δ 0 λ( 1 δ -t)p Q,1 (t)dt ) (25) + λ   Q-r-1 ∑ j=0 (λ(T -L)) j e -λ(T -L) j!   ( 2T - 1 δ + ∫ 1 δ 0 λ( 1 δ -t)p Q,1 (t)dt ) = λT   1 + Q-r-1 ∑ j=0 (λ(T -L)) j e -λ(T -L) j!   -Q -( λ δ -Q) Q-1 ∑ j=0 (λ/δ) j e -λ/δ j! + (λ/δ) Q e -λ/δ (Q -1)! .
To obtain T C(T, r, Q), it suffices to reuse Equation ( 23) by substituting the denominator by E(Cycle Length)

given in Equation ( 24). This finishes the characterization of the total cost for the two cases

1 δ + L ≤ T and 1 δ ≤ T < 1 δ + L.

Comparison

We illustrate the usefulness of the developed analytical results by describing numerical experiments that compare between the two cases of deterministic and exponential lifetimes. The cost parameters setting we consider is: We assume that the expected lifetime is 1 δ = 3 (time units). The cost parameters setting is similar to that in Chiu (1995b) and [START_REF] Berk | Analysis of the (Q,r) inventory model for perishables with positive lead times and lost sales[END_REF]. For each set of parameters, we compute the optimal review period, the optimal reorder point, the optimal order quantity and the optimal total operating cost, denoted by T * (time units), r * , Q * , and T C * (e per time unit), respectively. Note that in all the numerical experiments, we consider values for T * that satisfy the condition T * ≥ 1 δ .

K = {10,
Because of the complicated expressions of the steady state probabilities, it is unfortunately very hard to derive structural results for the total cost function under the two cases of deterministic and exponential lifetimes. Such results would be helpful for the computation of the optimal parameters T * , r * and Q * . From the numerical experiments, we observe in general that the total cost function is not unimodal and it may have more than one local extremum. This is consistent with the observations made in [START_REF] Springael | A lost sales inventory model with a compound poisson demand pattern[END_REF] and [START_REF] Hill | Optimal and near-optimal policies for lost sales inventory models with at most one replenishment order outstanding[END_REF]. We then resort to an exhaustive search algorithm over a broad range of (T, r, Q) values to numerically compute the optimal parameters.

The values of r and Q are discrete, 0 ≤ r ≤ Q and Q ≥ 1. To alleviate the computational efforts, we Table 3: Comparison between optimal costs for deterministic and exponential lifetimes (λ = 10, 1/δ = 3, L = 1, H = 1)

Cost parameters Deterministic lifetime Exponential lifetime restrict the optimization problem to discrete values of T in steps of 1. From all the obtained combinations for (T, r, Q), we finally choose the one leading to the best total cost, T C * . The results are given in Table 3.

T C * (Exp)-T C * (Det) T C * (Det) K C B W T * r * Q * T C * T * r * Q * T C
Table 3 reveals that the total cost with deterministic lifetimes is always lower than that with exponential lifetimes. This agrees with the well known property that performance deteriorates with variability.

For some combinations of the cost parameters the relative cost difference could be around 50%. This implies for managers that it is important to take the appropriate actions allowing to decrease the variability in lifetimes, by for example improving or standardizing the storage conditions of items.

We also observe from Table 3 that some cases yield to r * = 0 when the lifetime is exponentially distributed. This means replenishing only when the inventory is depleted. The observation agrees with the result in Theorem 3 of [START_REF] Weiss | Optimal ordering policies for continuous review perishable inventory models[END_REF]; because of the cost parameter values, it may be better to loose demand rather than to incur inventory holding and outdating costs. This observation suggests that managers should give a particular care in the estimation step of cost components. A wrong estimation would indeed lead to a poor inventory policy.

For our general model with Erlang lifetimes, we conjecture that the extreme cases of deterministic and exponential lifetimes are lower and upper bounds of the total cost, respectively. A rigorous proof of this result is too complex, but numerical illustrations using simulation are given in the next section.

The cost upper and lower bounds are useful for the system manager to get some sense on her actual cost. However, one would like to go beyond these two extreme points in order to deeper examine the effect of lifetime variability. This is unfortunately not possible using an analytical analysis. We then conduct simulation experiments in the next section to investigate to what extent lifetime variability deteriorates performance.

Simulation Experiments and Insights

In this section, we conduct a simulation study for the general case as described in Section 3, i.e., lifetimes follow an m-Erlang distribution with mean m/δ and m phases. When m = 1, the lifetime is exponential.

Since the coefficient of variation of the m-Erlang distribution is 1/ √ m, the lifetime distribution converges to the deterministic distribution with mean 1/δ as m -→ ∞. We therefore examine the impact of the lifetime randomness on the total cost, by varying the number of phases m.

We consider various sets in order to mix the effect of lifetime variability with the cost parameters.

The cost parameters setting is the same as that used in Section 4.3. We control the variability of lifetimes by varying the number of phases m from 1 (exponential distribution) to 10000 (nearly deterministic distribution). We use simulation-based optimization to compute the optimal policy parameters T * , r * and Q * . Simulation experiments run on Arena software, with the following sequence of events 1) An order arrives, 2) perished products are discarded, 3) demand is observed, 4) inventory Position is reviewed, 5)

an order is triggered.

Similarly to Section 4.3, we use an exhaustive search algorithm over a broad range of (T, r, Q) values to numerically compute the optimal parameters. The only difference here is that at each iteration, we use simulation to compute the total cost instead of using a numerical computation. The values of r and Q are discrete, 0 ≤ r ≤ Q and Q ≥ 1. To alleviate the computational efforts, we again restrict the optimization problem to discrete values of T in steps of 1. For a given set of parameters, we run a single simulation for a sufficiently large duration denoted by D. The duration D is chosen such that the performance measures do not vary for larger simulation durations, with a sufficiently high precision (four digits beyond the decimal point). For the considered settings, this is ensured with D = 20000 units of time. The performance measures are computed as follows. The expected inventory level per time unit, E(I), is computed as a time-persistent variable given by E D), where Γ(i, D) is the total time for which the inventory level is i. The expected rate of outdating items, E(O) is computed as the total number of outdating items divided by the simulation duration D. Similarly, the expected rate of lost sales, E(S), is computed as the total number of lost sales divided by D. Finally, the expected cycle length, E(Cycle Length), is computed as the expected value of all the realizations of the time durations between two successive orders. The results are given in Tables 4 and5, and in Figures 234567.

(I) = 1 D r+Q ∑ i=0 i • Γ(i,
A general conclusion from the numerical results is that the lifetime variability have an important effect on the total cost. The experiments illustrate the conjecture that the total cost of the extreme deterministic and exponential lifetime cases are lower and upper bounds of the total cost of the Erlang lifetime cases, respectively. When the coefficient of variation decreases from 1 to 0.44, the results indicate that the cost improvement ranges from 11% (for K = 50, C = 15, B = 20 and W = 5) to 46% (for K = 100, C = 5, B = 40 and W = 15). Thus, there might be opportunities for managers to considerably improve the operating cost by controlling the lifetime variability.

We observe that reducing lifetime variability has a diminishing return. Figures 2-5 reveal that most of the improvement in the total optimal cost takes place when moving from cases with high variability (coefficient of variation close to one). In other words, the lower is the variability, the lower is the sensitivity of the optimal cost. An important insight here is that an investment on reducing the item lifetime variability, through for instance the improvement of storage conditions, would be justified only for items with already non-negligible variability (larger than 0.1).

We also observe that the effect magnitude of the outdate cost W depends on the lifetime variability.

Figures 2345show that increasing the outdating cost significantly increases the total cost for only the cases with high lifetime variability. An explanation is as follows. To the contrary to the cases with low variability, the expected amount of perished products is important in the cases with high variability, and so is the corresponding cost component. Similarly, the performance deterioration with lifetime variability makes the total cost more sensitive to the lost sale cost parameter B in the case of high variability than in those with low variability. Moreover, Figures 6 and7 reveal that this sensitivity is further apparent in the case with high purchasing cost parameter C.

Finally, recall that the experiments shown so far are restricted to the case of a lead time that is lower than or equal to the period length, L ≤ T . Simulation experiments without this restriction are provided in Figures 8 and9. The experiments show that the conclusions found above still hold. 51 141.28 165.98 115.56 155.19 183.16 197.39 209.29 221.79 237.01 276.88 314.96 m = 5 87.63 107.34 131.89 98.50 118.06 142.26 179.45 198.82 212.27 204.74 224.45 249.10 m = 10 82.37 101.93 125.64 91.46 110.56 134.42 174.71 194.51 208.30 194.46 214.10 238.05 m = 50 77.94 96.86 118.83 84.50 103.30 125.96 171.02 189.63 203.54 185.10 203.98 227.31 W = 5 m = 100 77.42 95.97 117.53 83.68 102.37 124.38 170.64 188.84 203.07 183.94 202.70 225.62 m = 200 77.11 95.44 116.64 83.57 101.88 123.74 170.28 188.73 202.68 183.52 202.13 .16 120.77 88.13 106.69 129.69 171.10 189.47 203.75 186.80 205.35 228.54 m = 200 78.80 97.45 119.68 87.48 105.83 128.61 170.66 189.19 203.44 185.67 204.34 227.40 m = 500 78.78 97.39 119.52 87.43 105.57 128.16 170.36 188.87 203.28 185.47 203.95 227.02 m = 700 78.78 97.30 119.45 87.08 105.38 127.73 170.32 188.71 203.27 185.45 203.94 226.76 m = 1000 78.72 97.25 119.36 87.05 105.37 127.69 170.26 188.70 203.25 185.29 203.92 226.72 m = 10000 78.71 97.18 119.18 87.02 105.33 127.44 170.20 188.66 203.22 185.25 203.74 226.71 6 Conclusions and Future Research

We considered a (T, r, Q) lost sales inventory system with perishable items. Item lifetimes are random and assumed to follow an Erlang distribution. This is a rich modeling allowing to cover various cases of lifetime variability in practice. For the extreme cases of exponential and deterministic lifetimes, we provided an analytical approach for the system performance evaluation during the steady-state regime, mainly the expected inventory level, the rates of orders, perished items and lost sales. This allowed to obtain some sense on the effect of the lifetime variability on the total optimal cost. A more comprehensive analysis for the general case is then conducted using simulation. We have examined the important effect of lifetime variability mixed with the cost parameters on the total operating cost. It appeared that there might be opportunities for managers to considerably improve the operating cost in the case of high lifetime variability. This is more apparent for cases with large outdate, lost sale, and purchasing cost parameters.

In a future research, it would be useful to extend the analytical analysis to the Erlang distributed lifetimes. Another interesting work is to prove the stochastic ordering between systems with different lifetime variabilities. An ambitious future work would be to extend the model we developed herein to the case of one warehouse multiple retailers inventory systems for perishables. We believe that the work of [START_REF] Shang | Optimal and heuristic echelon (r, nQ,T) policies in serial inventory systems with fixed costs[END_REF] could be extended to the case of perishables with random lifetime and insights regarding the impact of perishability on the optimal policy could be examined.

Figure 1 :

 1 Figure 1: The (T, r, Q) periodic review inventory policy

  Probability of Inventory Levels. Consider the discrete-time stochastic process {I n = I(nT + L), n ≥ 0} which describes the inventory level on-hand at the specific epochs nT + L for n ≥ 0. It is easy to see that it is a discrete-time Markov chain. Moreover, {I n , n ≥ 0} is finite, aperiodic and irreducible. Then its steady-state probabilities, P (i) = lim n→∞ Pr{I n = i}, exist and are unique, for 0 ≤ i ≤ r + Q. They are the solutions of the fixed-point equation P = P • U • V and the normalization equation, where P = {P (i), 0 ≤ i ≤ r + Q} is the vector of the equilibrium probabilities, and U and V are transition matrixes of order (r + Q + 1). They are given by

  100, 150} (e per order), C = {5, 15} (e per item), W = {5, 15} (e per perished item), B = {20, 40} (e per lost customer) and H = 1 (e per stored item per time unit). The constant lead time is L = 1 (time unit) and the expected arrival rate of customers is λ = 10 (per time unit).

Figure 2 :Figure 3 :

 23 Figure 2: Optimal total cost for C=5, B=20 and H=1 (λ = 10, 1/δ = 3, L = 1) Figure 3: Optimal total cost for C=5, B=40 and H=1 (λ = 10, 1/δ = 3, L = 1)

Table 1 :

 1 Modeling features in key papers on perishable inventory systems

	Lead time distribution Lifetime distribution Demand distribution
	Replenishment policy Planning horizon Type of review Excess demand
	Article

Table 2 :

 2 List of Notations

	λ :	Mean demand rate
	1/δ :	Mean lifetime of products
	m :	Number of phases for the Erlang distributed lifetimes
	L :	Replenishment lead time (constant)
	K :	Fixed ordering cost per order
	H :	Holding cost per unit of product held in stock per unit of time
	C :	Purchase cost per unit of product
	B :	Lost sales cost per unit of product
	W :	Outdate cost per unit of product that perishes in stock
	E(O) :	Expected rate of outdating items
	E(S) :	Expected rate of lost sales
	E(I) :

Table 4 :

 4 Optimal inventory policy (λ = 10, 1/δ = 3, L = 1, H = 1)

	Cost Parameters				Lifetime distribution, m = 1				Lifetime distribution, m = 50
	K	C B W T * r * Q * E(I) E(O) E(S) E(Cycle Length)	T C *	T * r * Q * E(I) E(O) E(S) E(Cycle Length)	T C *
	10	5 20 5	1 22 11 6.36	2.12	1.13	1.00	104.51 1 21 20 13.70 0.30	0.42	2.03	77.94
	50	5 20 5	1 17 24 10.24 3.41	1.47	2.01	141.28 1 18 22 14.14 0.39	0.43	2.22	96.86
	100 5 20 5	1 14 25 10.47 3.49	1.52	2.09	165.98 1 16 24 13.89 0.44	0.66	2.46	118.83
	10	5 40 5	1 26 13 9.86	3.29	0.37	1.01	115.56 1 22 21 15.69 0.51	0.26	2.05	84.50
	50	5 40 5	1 24 13 9.61	3.20	0.41	1.02	155.19 1 20 22 15.26 0.50	0.29	2.16	103.30
	100 5 40 5	1 22 29 14.75 4.92	0.56	2.02	183.16 1 19 24 15.94 0.63	0.31	2.33	125.96
	10	5 20 15	1 19 10 5.07	1.69	1.70	1.00	124.32 1 18 20 12.39 0.21	0.59	2.08	80.30
	50	5 20 15	1 15 10 4.94	1.65	1.80	1.02	164.10 1 16 20 11.59 0.17	0.74	2.13	99.47
	100 5 20 15	1	9 18 6.01	2.00	3.21	2.05	193.12 1 15 21 11.83 0.20	0.79	2.24	122.28
	10	5 40 15	1 24 12 7.95	2.65	0.68	1.00	144.82 1 21 21 15.07 0.45	0.29	2.07	89.07
	50	5 40 15	1 22 12 7.84	2.61	0.71	1.01	184.59 1 20 22 15.26 0.50	0.29	2.16	108.25
	100 5 40 15	1 19 26 11.91 3.97	1.05	2.01	227.62 1 20 22 15.26 0.50	0.29	2.16	131.38
	10 15 20 5	1 10 6	1.86	0.62	4.62	1.00	197.39 1 18 18 9.94	0.08	1.07	2.01	171.02
	50 15 20 5	4	0	6	0.46	0.15	8.65	4.00	209.29 1 11 20 9.15	0.08	1.71	2.40	189.63
	100 15 20 5	4	0	6	0.46	0.15	8.65	4.00	221.79 3	0 23 5.34	0.08	5.57	5.16	203.54
	10 15 40 5	1 24 11 6.37	2.12	1.13	1.00	237.01 1 21 20 13.70 0.30	0.42	2.03	185.10
	50 15 40 5	1 19 12 7.42	2.47	0.86	1.03	276.88 1 19 21 13.90 0.35	0.41	2.12	203.98
	100 15 40 5	1 16 24 10.18 3.39	1.49	2.02	314.96 1 18 22 14.14 0.39	0.43	2.22	227.31
	10 15 20 15	4	0	4	0.23	0.08	9.08	4.00	200.38 2 17 17 8.44	0.04	1.53	2.01	171.63
	50 15 20 15	4	0	4	0.23	0.08	9.08	4.00	210.38 1 10 20 8.63	0.07	2.03	2.50	190.34
	100 15 20 15	4	0	4	0.23	0.08	9.08	4.00	222.88 3	0 23 5.34	0.08	5.57	5.16	204.37
	10 15 40 15	1 19 11 6.27	2.09	1.17	1.01	258.13 1 19 20 12.90 0.25	0.53	2.06	188.03
	50 15 40 15	1 17 11 6.09	2.03	1.27	1.02	297.68 1 19 20 12.90 0.25	0.53	2.06	207.44
	100 15 40 15	2 18 20 7.42	2.47	2.47	2.00	343.43 1 17 21 12.84 0.26	0.56	2.17	230.75
						Lifetime distribution, m = 500				Lifetime distribution, m = 10000
	K	C B W T	r	Q E(I) E(O) E(S) E(Cycle Length)	T C *	T	r	Q E(I) E(O) E(S) E(Cycle Length)	T C *
	10	5 20 5	1 18 21 13.39 0.20	0.47	2.16	76.98	1 19 21 13.84 0.24	0.41	2.15	76.83
	50	5 20 5	1 18 22 14.16 0.29	0.39	2.22	95.30	1 18 22 14.19 0.28	0.39	2.23	95.08
	100 5 20 5	3 24 30 15.37 0.71	0.70	3.00	116.22 3 24 30 15.53 0.70	0.68	3.00	116.05
	10	5 40 5	1 21 22 15.84 0.45	0.24	2.16	83.41	1 21 22 15.68 0.43	0.24	2.16	83.05
	50	5 40 5	1 21 23 16.46 0.55	0.20	2.23	101.30 1 20 23 15.89 0.47	0.24	2.25	101.07
	100 5 40 5	1 21 24 17.19 0.66	0.17	2.29	123.26 1 20 24 16.63 0.60	0.20	2.31	122.81
	10	5 20 15	1 18 20 12.35 0.14	0.60	2.10	78.78	1 17 20 11.97 0.12	0.65	2.12	78.71
	50	5 20 15	1 16 21 12.41 0.15	0.63	2.21	97.39	1 17 21 12.87 0.17	0.54	2.18	97.18
	100 5 20 15	1 15 22 12.53 0.20	0.71	2.34	119.52 1 15 23 12.99 0.23	0.71	2.43	119.18
	10	5 40 15	1 19 22 14.60 0.33	0.34	2.21	87.43	1 20 22 15.13 0.36	0.29	2.19	87.02
	50	5 40 15	1 19 22 14.60 0.33	0.34	2.21	105.57 1 20 22 15.13 0.36	0.29	2.19	105.33
	100 5 40 15	1 18 23 14.80 0.39	0.34	2.29	128.16 1 19 23 15.36 0.42	0.28	2.27	127.44
	10 15 20 5	1 14 18 9.22	0.02	1.27	2.07	170.08 1 14 18 9.20	0.02	1.27	2.07	169.95
	50 15 20 5	1 11 20 9.18	0.03	1.67	2.40	188.38 1 11 23 10.29 0.10	1.73	2.77	188.18
	100 15 20 5	3	0 25 5.84	0.08	5.54	5.56	202.55 3	0 25 5.83	0.08	5.54	5.56	202.47
	10 15 40 5	1 20 21 14.25 0.28	0.38	2.13	183.44 1 19 21 13.84 0.24	0.41	2.15	182.92
	50 15 40 5	1 19 22 14.68 0.33	0.33	2.21	201.87 1 18 22 14.19 0.28	0.39	2.23	201.46
	100 15 40 5	1 18 22 14.16 0.29	0.39	2.22	224.47 1 18 22 14.19 0.28	0.39	2.23	223.88
	10 15 20 15	1 14 18 9.23	0.02	1.27	2.07	170.36 1 14 18 9.20	0.02	1.27	2.07	170.20
	50 15 20 15	1 10 20 8.63	0.03	2.03	2.52	188.87 1 11 20 9.19	0.03	1.68	2.41	188.66
	100 15 20 15	3	0 23 5.37	0.04	5.54	5.15	203.28 3	0 23 5.37	0.04	5.54	5.15	203.22
	10 15 40 15	1 18 21 13.39 0.20	0.47	2.16	185.47 1 18 21 13.42 0.21	0.46	2.16	185.25
	50 15 40 15	1 18 21 13.39 0.20	0.47	2.16	203.95 1 18 21 13.42 0.21	0.46	2.16	203.74
	100 15 40 15	1 18 21 13.30 0.21	0.47	2.17	227.02 1 17 22 13.65 0.24	0.47	2.26	226.71
									20

  224.57 m = 500 76.98 95.30 116.22 83.41 101.30 123.26 170.08 188.38 202.55 183.44 201.87 224.47 m = 700 76.91 95.24 116.14 83.33 101.22 123.03 170.04 188.35 202.51 183.28 201.83 224.27 m = 1000 76.86 95.20 116.09 83.09 101.16 122.87 170.03 188.25 202.48 183.03 201.55 224.15 m = 10000 76.83 95.08 116.05 83.05 101.07 122.81 169.95 188.18 202.47 182.92 201.46 223.88 m = 1 124.32 164.10 193.12 144.82 184.59 227.62 200.38 210.38 222.88 258.13 297.68 343.43 m = 5 96.69 116.51 141.01 111.34 130.74 154.99 181.58 200.38 213.66 214.35 234.17 258.95

		m = 10	87.67 107.47 131.67 99.33 118.91 143.39 176.05 196.01 210.08 200.20 219.99 244.69
		m = 50	80.30	99.47 122.28 89.07 108.25 131.38 171.63 190.34 204.37 188.03 207.44 230.75
	W = 15	m = 100	79.40	98


Karim Ghanes

Oualid Jouini

Mathias Wargon

Zied Jemai

Modeling and analysis of triage nurse ordering in emergency departments

Keywords: emergency departments, triage nurse ordering, simulation, length of stay I

come

INTRODUCTION

Emergency Department (ED) is the service within hospitals responsible for providing care to life-threatening and other emergency cases over 24 hours daily, 7 days a week. Nowadays, EDs are facing a recurrent worldwide crisis, namely overcrowding. Using various methods, healthcare practitioners and researchers in operations management try to alleviate this overcrowding.

In order to improve ED performance, an extensive body of the literature consists on introducing additional resources (physicians, beds, etc.). This includes either resource allocation issues [START_REF] Duguay | Modeling and Improving Emergency Department Systems Using Discrete Event Simulation[END_REF][START_REF] Bucheli | Reduced length of stay in medical emergency department patients: a prospective controlled study on emergency physician staffing[END_REF][START_REF] Rossetti | Emergency department simulation and determination of physician staffing schedules[END_REF] or process modification issues [START_REF] Oredsson | A systematic review of triage-related interventions to improve patient flow in emergency departments[END_REF] such as some interventions in triage (team triage and physician at triage), point of care testing, etc. This is the most direct way to alleviate crowding and improve responsiveness. However, as reported in [START_REF] Saghafian | Patient Streaming as a Mechanism for Improving Responsiveness in Emergency Departments[END_REF], because it is also the most expensive approach, it is generally not the preferred option. High salaries of doctors and high costs of medical equipments [START_REF] Sinreich | Emergency Department Operations: The Basis for Developing a Simulation Tool[END_REF][START_REF] Warner | Personnel staffing and scheduling, in Patient Flow: Reducing Delay in Healthcare Delivery[END_REF] combined to budgetary restrictions has prompted practitioners and researchers to investigate new methods that are costeffective. Among these, Triage Nurse Ordering (TNO) appears to be a promising approach that does not require any resource investment. It can be achieved using existing staff with little additional training [START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF].

TNO is an advanced triage intervention that consists on allowing triage nurse to order tests and treatments before the patient is seen by the physician [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF][START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF][START_REF] Pallin | Mercy Hospital: simulation techniques for ER processes[END_REF]. The common protocol in the ED is that triage nurse cannot order diagnosis tests. He/she is essentially responsible of making a first assessment of patients' state and categorizing them into different acuity levels. The decision of requiring diagnosis tests or not is traditionally under the responsibility of the ED physician. However, the medical literature suggests that with an appropriate education and training, and adapted protocoled guidelines, triage nurses could be able to order some tests to a level comparable to that of physicians [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF][START_REF] Free | Literature review of studies on the effectiveness of nurses ability to order and interpret Xrays[END_REF][START_REF] Fry | Triage nurses order x-rays for patients with isolated distal limb injuries: a 12-month ED study[END_REF][START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF]. Diagnostic imaging and laboratory tests are time-consuming processes in the ED that are associated with longest length of stay (LOS) [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF][START_REF] Vegting | Analysing completion times in an academic emergency department: coordination of care is the weakest link[END_REF][START_REF] Yoon | Analysis of factors influencing length of stay in the emergency department[END_REF]. If tests are early requested in the triage process, they could be undergone without waiting for the first examination by the ED physician, and test results could be reviewed by the latter as soon as he becomes available.

TNO has been related to enhanced patient satisfaction [START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF][START_REF] Parris | Do triage nurse initiated x-rays for limb injuries reduce patient transit time?[END_REF][START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF]. However, "little is known about the effectiveness of this intervention in improving ED time metrics" [START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF]. Only few medical papers reported that TNO could possibly reduce the ED LOS [START_REF] Retezar | The effect of triage diagnostic standing orders on emergency department treatment time[END_REF][START_REF] Cheung | An advance triage system[END_REF][START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF][START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF]. As mentioned in [START_REF] Oredsson | A systematic review of triage-related interventions to improve patient flow in emergency departments[END_REF], there is only limited scientific evidence that having nurses to request certain tests results in shorter waiting time and LOS. Moreover, as highlighted by [START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF], the existing LOS improvements revealed in the literature may range widely (from 2.45 to 74 minutes). There is a real need to conduct studies that will legitimize the use of TNO in EDS in terms of LOS reduction [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF][START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF].

In the present paper, the objective is to analyze the effect of TNO on ED time metrics taking into consideration the key parameters of such an intervention. We relied on an online survey that we performed with EDs from different countries in order to understand the current practices and obtain specialists' opinions. The survey's questions focused on the relevancy of TNO, on which types of diagnosis tests exactly it could be applied and why. This survey helped us to understand the whys and wherefores of this problem in order to delimit the framework and include the most relevant parameters related to TNO in our model. Using simulation, we assess the effectiveness of TNO as a function of triage nurse ability level, and investigate other elements that could have an impact on this effectiveness (System load, service time prolongation). We derive useful insights that can assist decision makers when implementing a TNO intervention.

The rest of the paper is organized as follows. A literature review on TNO is presented in section 2. In Section 3, we report the results of the performed survey and describe in detail our TNO model. In Section 4, we conduct experiments. Finally, in section 5, we summarize the main insights and highlight some future research.

II. LITERATURE REVIEW

TNO, also called advanced triage [START_REF] Rosmulder | Advanced triage improves patient flow in the emergency department without affecting the quality of care[END_REF][START_REF] Cheung | An advance triage system[END_REF], is a worldwide ED issue that was addressed in North America [START_REF] Retezar | The effect of triage diagnostic standing orders on emergency department treatment time[END_REF][START_REF] Cheung | An advance triage system[END_REF], Europe [START_REF] Rosmulder | Advanced triage improves patient flow in the emergency department without affecting the quality of care[END_REF][START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF], Asia [START_REF] Ching | Initiation of x-rays by the triage nurse: competency and its effect on patients' total time spent in the accident and emergency department[END_REF][START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF] and Australia [START_REF] Parris | Do triage nurse initiated x-rays for limb injuries reduce patient transit time?[END_REF]. The prior literature examining TNO is almost exclusively addressed from a medical perspective. It consists in general on 2 types of empirical studies: examining the ability of triage nurses in initiating diagnosis tests properly [START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF], assessing the effect of such an intervention on ED time metrics [START_REF] Cheung | An advance triage system[END_REF][START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF][START_REF] Parris | Do triage nurse initiated x-rays for limb injuries reduce patient transit time?[END_REF], or both [START_REF] Rosmulder | Advanced triage improves patient flow in the emergency department without affecting the quality of care[END_REF][START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF]. For the first type, the method consists generally on using the attending physician as a standard to judge the accuracy of triage nurse orders, and for the second it consists on comparing statistically two samples of patients (one with the traditional ED process and another with TNO) in terms of time metrics, mainly LOS.

As highlighted in [START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF] and [START_REF] Oredsson | A systematic review of triage-related interventions to improve patient flow in emergency departments[END_REF], most of TNO interventions are limited to some radiographs, mostly joints and bones of distal limbs [START_REF] Fry | Triage nurses order x-rays for patients with isolated distal limb injuries: a 12-month ED study[END_REF][START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF][START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF]. However, some include additional diagnostic test requesting such as blood tests, urinalysis, electrocardiogram [START_REF] Cheung | An advance triage system[END_REF][START_REF] Winn | Emergency Department Efficiency Through Utilization of Triage Nurse Protocols[END_REF][START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF][START_REF] Kirtland | Simulating an emergency department 'is asmuch fun as[END_REF] and radiographs of other parts like the skull [START_REF] Ching | Initiation of x-rays by the triage nurse: competency and its effect on patients' total time spent in the accident and emergency department[END_REF]. Even if there is some unanimity about distal limb radiographs, the choice of the diagnosis type for TNO is rarely justified in the literature and still remains unclear.

TNO has been related in some papers to a decreased LOS. Reference [START_REF] Rosmulder | Advanced triage improves patient flow in the emergency department without affecting the quality of care[END_REF] reported that LOS decreased by 27 minutes (18%) with foot/ankle X-rays initiated at triage. In [START_REF] Cheung | An advance triage system[END_REF], time savings was on average 46 min in the total LOS with TNO applied on some X-rays and blood tests. In [START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF], total LOS for patients with radiographs requested by the nurse was on average 18.59 minutes less than the overall average. Reference [START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF] reported that a mean reduction of 37.2 min (36%) from time of triage to time of treatment decision was achieved in the group of patients with triage initiated X-rays comparing to control group. However, according to [START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF] and [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF], there is a paucity of research examining the effect of TNO intervention on ED time metrics. Time reductions related to TNO may range widely (from 2.45 to 74 minutes according to [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF]) and some negative conclusions have also been reported. Reference [START_REF] Parris | Do triage nurse initiated x-rays for limb injuries reduce patient transit time?[END_REF] performed a comparison between a group of patients who had X-ray initiated in triage and a group with a regular pathway, and found that the difference in LOS between the two groups was not statistically significant. However, staff and patient satisfaction with this change was high and justified the continuation of the practice in the ED.

Satisfaction was not formally measured but it was reported that physician's satisfaction was increased through the availability of diagnostic results since the first examination; patients seemed satisfied for using the waiting time more efficiently in addition to a greater sense of team working for all staff [START_REF] Cheung | An advance triage system[END_REF][START_REF] Fry | Triage nurses order x-rays for patients with isolated distal limb injuries: a 12-month ED study[END_REF][START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF][START_REF] Parris | Do triage nurse initiated x-rays for limb injuries reduce patient transit time?[END_REF].

One of the arguments facing TNO is the fear of overrequesting diagnostic tests that would not have been ordered by the physician [START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF]. The potential benefits of TNO in terms of time savings and satisfaction must be balanced with the disadvantages of such excessive requests: additional time, additional expense and increased resource utilization, unnecessary radiation exposure and potential morbidity [START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF][START_REF] Mcarthur | Comparison of triage nurse versus emergency physician ordering of extremity radiographs[END_REF]. For instance, [START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF] addressed the problem of triage nurse ability in ordering radiographs including 934 patients in their study. The triage nurse requested radiographs for 94.54% of patients (883). 5.44% of requested tests (48 out of the 883) were considered unnecessary by the case physician.

Under-requesting is another type of possible error. Triage nurse could miss some necessary tests that will further be required by the physician. In [START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF], among the 51 cases where triage nurse did not order any radio (5.5%), 65% (33/51) had an X-ray requested by the attending physician (underrequesting rate of 3.5 %, 33/934). TNO presents also a risk of additional tests following the physician's examination. Additional views of the same/adjacent or different regions can be ordered because the first view did not demonstrate the problem or another injury was discovered during the consultation [START_REF] Oredsson | A systematic review of triage-related interventions to improve patient flow in emergency departments[END_REF][START_REF] Macleod | Should nurses be allowed to request X-rays in an accident & emergency department?[END_REF]. "Additional trips to the radiology department become necessary, increasing both the time required for treatment and the inconvenience to the patient" [START_REF] Mcarthur | Comparison of triage nurse versus emergency physician ordering of extremity radiographs[END_REF]. In [START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF], 11% of ordered radios (97 out of 883) were followed by additional ones after physician's assessment.

In addition to [START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF], a few other papers examined the ability of triage nurses in initiating radiographs appropriately (using similar inclusion criteria). The reported statistics from most complete studies are provided in table I. [START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF] Note that when additional tests are ordered by the physician, studies did not mention whether the ones ordered by the nurse were necessary or not. No statistics were formally reported about patients with both over-requesting and underrequesting (additional tests after unnecessary tests) except in rare papers like [START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF] where these patients represented 15%.

The success of TNO can likely be achieved using existing triage nurses with little additional training [START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF]. In most of the analyzed papers, triage nurses' skills were extended before experiments with training programs on examination skills and inclusion/exclusion criteria for exams requisition [START_REF] Cheung | An advance triage system[END_REF][START_REF] Fry | Triage nurses order x-rays for patients with isolated distal limb injuries: a 12-month ED study[END_REF][START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF]. As demonstrated by [START_REF] Seaberg | Correlation between triage nurse and physician ordering of ED tests[END_REF], the ability of triage nurse in ordering tests can be improved with the use of test ordering guidelines. Reference [START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF] reports a reduced gap between triage nurse and physician ability in ordering radiographs after participating in a 1-day training program and by using carefully designed protocols. However, there are no standardized guidelines for TNO interventions [START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF]. Reported trainings are various in time and contents [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF][START_REF] Rowe | The role of triage nurse ordering on mitigating overcrowding in emergency departments: a systematic review[END_REF]. In addition to an initial non-uniformity in nurse education between countries, and also within the same country [START_REF] Free | Literature review of studies on the effectiveness of nurses ability to order and interpret Xrays[END_REF], different TNO trainings and protocols could have influenced results reported in studies [START_REF] Robinson | An integrative review: Triage protocols and the effect on ED length of stay[END_REF][START_REF] Thurston | Should accident and emergency nurses request radiographs? Results of a multicentre evaluation[END_REF].

As explained above, TNO contributions consisted either on measuring the impact of such an intervention on time metrics or assessing triage nurse ability. However, it should be noted that no paper analyzed or quantified the impact of nurse ability on patient time in the ED so far. Service times' prolongation is also an element that was not addressed. As mentioned in [START_REF] Lindley-Jones | Triage nurse requested xrays: are they worthwhile?[END_REF] and [START_REF] Lee | Accuracy and efficiency of x-ray requests initiated by triage nurses in an accident emergency department[END_REF], the average time of triage and consultation could be lengthened under TNO which could also affect the results.

In the OR/OM domain, we identify very limited contributions. Reference [START_REF] Kirtland | Simulating an emergency department 'is asmuch fun as[END_REF] uses simulation to test several staffing and process alternatives in order to reduce the patients LOS in an ED (TNO, fast track, point of care testing, etc.). There was no significant time savings related to TNO (3.6 minutes on the total average LOS). However, authors suggested that TNO would be more effective when the system is quite busy, but that was not demonstrated. With the exception of the considered types of exams (X-rays, lab tests and ECG), no information was provided about the parameters included in the TNO model (changes in the patient pathway, triage nurse ability, etc.).

III. MODEL DEFINITION

In order to understand specialists' opinions about TNO and define the appropriate model framework, a survey was performed in many EDs in France, USA, Netherlands, Germany, Belgium, Greece and Tunisia.

A. Survey results

The results are based on 36 practitioners (ED managers and ED physicians) from 28 different EDs. 75% of the answers were provided by specialists from French EDs. The majority of surveyed specialists (86%) considered TNO as a potential relevant practice in general. However as shown in table II, the feasibility of TNO varies greatly from one test type to another. For each one, the experts provided the practical reasons about the possibility to apply TNO or not. Conventional radiology, also called standard radiology, radiographs or X-rays, were considered by the experts as the most appropriate diagnosis tests for a TNO intervention in their EDs. Only some particular types of radiographs are concerned which are those for simple extremity traumatology of stable patients. That includes bones and joints radiographs of distal limbs that are below the large joints like the hip and the shoulder (hand, wrist, elbow, foot, knee, ankle, etc.). They are routine tests, easy to perform with limited risk for patients (noninvasive). Multi trauma cases as well as radiographs of other parts such as spine, chest, abdomen and pelvis should be excluded.

For many reasons, CT-scan, MRI and Echo were judged inappropriate for a TNO intervention. They are much more expensive and represent critical resources in the hospital. The application of TNO on these tests requires a medico-economic evaluation. Moreover, they are invasive and more specific tests. The decision of ordering such tests is complex and cannot be done without a complete (physical and clinical) examination by a physician.

Biological tests are also complex and costly and the opinion of experts about the application of TNO on them is mixed. The survey revealed 83% of favorable views for TNO of urinalysis. However, that was limited by specialists to certain basic urine tests. The mainly mentioned candidate is a type of urine analysis called urine test strip. It is a basic and quick diagnosis tool that is used by practitioners in the ED without even resorting to the laboratory. Other kinds of basic urine tests were mentioned such as urine pregnancy tests. According to the survey, blood tests can be ordered by triage nurse only in certain cases (fever in a patient back from a tropical country or in a patient receiving chemo, diabetes, HIV testing, etc.). More specific and sophisticated blood tests require a clinical examination by a physician and must be discussed on a caseby-case basis. The disparity in the situations requiring biological testing makes them difficult to generalize.

There is unanimity on allowing triage nurse to decide about an Electrocardiogram (ECG) in several cases like chest pain. This protocol is more common than other tests and is already applied in many surveyed EDs.

B. The model description

Given the answers collected from the performed survey, the study will focus on low acuity level patients (ESI4 and ESI5) with distal limbs traumatology requiring conventional radiographs (X-rays). In France, Trauma injuries represent about half of ED visits [START_REF] Potel | L'organisation de l'aval des urgences: état des lieux et propositions[END_REF][START_REF] Carrasco | Les usagers des urgences : premiers résultats d'une enquête nationale[END_REF]. Among these, trauma to the extremity whether upper like wrist and hand injuries, or lower like ankle sprain represent the most common cases in EDs particularly among non-critical patients, with X-rays being the reference test [START_REF] Ganansia | Quand ne pas prescrire de radio en traumatologie des extrémités chez l'adulte[END_REF].

For this kind of patients, length of stay (LOS) is the key metric [START_REF] Ghanes | Key performance indicators for emergency departments: A survey from an operations management perspective[END_REF][START_REF] Saghafian | Patient Streaming as a Mechanism for Improving Responsiveness in Emergency Departments[END_REF]. "Due to relatively minor nature of these injuries, those patients have often to wait a long time for treatment and investigation in EDs" [START_REF] Parris | Do triage nurse initiated x-rays for limb injuries reduce patient transit time?[END_REF]. This group of patients rarely requires biological tests which make them free of any necessary sampling, and allow sending them to radiology right after triage.

The interviews with experts combined with the existing literature allowed to identify the most relevant parameters that could have an impact on the effectiveness of TNO, and that will further constitute the basis of our experiments:

-The accuracy of triage nurse in requesting tests (over, under and incomplete-requesting) -The ED level of crowding -The impact of TNO on some service times As shown in Fig. 1, comparing to the traditional patient pathway, when triage nurse orders diagnosis tests, they are initiated right after triage. Since ordering tests is an additional task for triage nurse, the triage service time represented in our model by a random variable T could be increased by a certain amount of time ∆T. When tests are completed, the physician examines the patient for the first time and interprets his/her tests results during one single aggregated consultation. In our model, we make the assumption that this task has the same time distribution with a regular consultation. The TNO path depicted in Fig. 1 is a simplified representation that corresponds to a particular ideal scenario under TNO (see N+/P+ in Fig. 2). According to the case and the triage nurse risk of error, we distinguish in total between six possible situations. The latters are depicted with the appropriate formalism in Fig. 2. (N+/P++) is a particular situation of under-requesting where triage nurse orders some necessary tests while missing others (the physician may order radiographs of other parts or different views of the same), which imposes additional trips to radiology. (N+/P-+) is a combination of both under and over requesting in which the patient is first sent unnecessarily to radiology by triage nurse, and then sent back later by the physician. In addition to potential loss of time, these situations have an impact on patient's convenience and satisfaction.

As mentioned in the literature review, in empirical studies, reported statistics on the two situations (N+/P++) and (N+/P-+) are merged. Thus, they will also not be differentiated in our model and will all be considered as (N+/P++). In other words, we make the assumption that when the physician orders additional tests, the ones ordered by triage nurse are never completely useless. Consequently, we obtain five possible patient paths under TNO that are represented in Fig. 3.

IV. EXPERIMENTS

Our experiments consist on a case study involving a French ED (Saint-Camille) and are divided into 4 parts. In the first one, the objective is to understand the impact of nurse's abilities and decisions on TNO effectiveness. We calculate the expected improvement in LOS as a function of a realistic range of TNO-related probabilities. In the second part, we extend the analysis by varying the different probabilities within a wider range of values in order to figure out what is the most harmful nurse error. In the third part, we assess the relationship between the ED load and TNO effectiveness. Finally, we assess the impact of triage processing time prolongation on the system performance.

Collected data from Saint Camille indicate that eligible patients represent 17% of the total number of patients. A realistic discrete event simulation model is used to perform our experiments on Saint Camille ED [START_REF] Ghanes | Simulation-based optimization of staffing levels in an emergency department[END_REF]. In this model, all common structural and functional characteristics of EDs, at least in France, were taken into consideration thanks to a close collaboration with practitioners.

For each simulation, we use one replication of 100 weeks with a warm-up period of one week. For each rate, we use the lowest and the highest value found in the literature (see table 1) and generate all the possible combinations (2 4 =16 in total). For instance, the scenario H-LLH refers to a TNO intervention where triage nurse has a high requesting rate, a low probability of over-requesting, a low probability of under-requesting and a high probability to require incomplete tests (that will be followed by a physician test order). The LOS improvement of each scenario for eligible patients is depicted in Fig. 4 and Fig. 5. The derived insights can be summarized as follows:

-Within the used ranges of trained triage nurses' ability reported by empirical studies, TNO is a beneficial intervention for all combinations. -For any given trained nurse ability (for any set of over/under/incomplete requesting rate), trained triage nurse should preferably order tests as much as possible while respecting the predetermined protocols. -Under-requesting appears to be harmless. Over and incomplete requesting both reduces the benefit from TNO.

The worst scenarios are the situations L-HHH and L-HLH (19.31% among eligible patients and 4.49% of overall improvement) where triage nurse has the worst abilities while having a low requesting rate. The best scenario is the opposite situations H-LLL and H-LHL (29.38% among eligible patients and 7.65% overall).

B. Extended analysis of TNO effectiveness as a function of the key probabilities

So far, no conclusion can be drawn about which triage nurse's ordering error is the most harmful. This is because the used error probabilities coming from the literature are limited. They correspond to the ability of triage nurses that were preliminarily trained. In what follows we will experiment the best scenario H-LLL by varying one by one each probability rate from 0% to 100%. The following insights can be derived:

-The benefit from TNO is more apparent for higher requesting rates (with respect to protocols). -The risk of under-requesting rate has no impact on TNO performance -The risk of over-requesting and the risk of incomplete requesting (additional tests further requested by physician) affect TNO performance and have similar impacts on it. This result is quite intuitive since both of them consist of an additional trip to radiology department. This result holds under the assumption that when the physician orders additional tests, the ones ordered by triage nurse were not completely useless. Otherwise, the rate of additional tests would be the most harmful. -For over and incomplete requesting rates, there is a threshold under which TNO could be detrimental for the system performance.

C. TNO effectiveness as a function of the system load

The actual arrival pattern in Saint-Camille ED depends on the day of the week and the hour of the day. Similarly to [START_REF] Yom-Tov | Erlang-r: A time-varying queue with reentrant customers, in support of healthcare staffing[END_REF] and [START_REF] Ahmed | Simulation optimization for an emergency department healthcare unit in Kuwait[END_REF], we assume that arrivals follow a non-homogenous Poisson process (7x24 arrival rates).

In order to assess the relationship between the system workload and the expected benefit from TNO, we perform a sensitivity analysis by varying arrival rates, in the best and the worst scenarios of Section A (see Fig. 7). The following conclusion can be drawn: The benefit derived from TNO is more apparent for heavily loaded EDs.

D. The impact of triage service time extension on TNO effectiveness

In what follows we address the question of triage service time prolongation because of TNO and assess its impact on TNO effectiveness. According to data collection (experts' judgment in particular), the distribution of triage service time is assumed to be Normal (7, 1.5). Using the best and the worst scenarios from part 1 (H-LLL and L-HHH respectively), we perform a sensitivity analysis on triage time by extending it up to 200% (see Fig. 8). -Triage time prolongation lowers TNO effectiveness, but should not be considered as a major concern for reasonable time prolongations and nurse abilities. -Triage time prolongation can make TNO detrimental for the overall system. For instance, for limited triage nurse abilities (L-HHH) and high triage prolongations (from 150%), TNO remains beneficial for eligible patients but the overall system LOS is affected by longer waiting and processing times for triage.

V. CONCLUSION

The present paper represents the interface and the link between the two commonly addressed TNO issues, namely the assessment of triage nurse ability and the assessment of the effectiveness of this intervention in improving ED time metrics. We derived useful insights that will help decision makers for a successful implementation of TNO. For instance, TNO is always beneficial within a reasonable range of triage nurse ability level. However, there is a threshold on this ability under which TNO could be detrimental for the system performance. The benefit derived from TNO is more apparent for heavily loaded EDs. We also demonstrate that triage time prolongation does not have a significant impact on eligible patients but can affect the rest of the ED patients and the overall ED performance. Although the modeling is based on a specific ED, qualitative conclusions hold for other ED frameworks. As a perspective, an analytical modeling for TNO would be helpful for the generalization of the aforementioned results.

FIGURE 1 :

 1 FIGURE 1: REGULAR AND TNO PATIENT PATHWAY

FIGURE 2 :

 2 FIGURE 2: POSSIBLE SITUATIONS UNDER TNO (N+/P+) is the ideal scenario that embodies the utility of TNO. Triage nurse orders the appropriate tests which would allow saving the time of a first consultation and its corresponding waiting times (WTs). In scenario (N-/P-), the nurse is right and this situation has neither advantages nor disadvantages. (N-/P+) is a harmless situation of underrequesting where the TNO patient pathway is unintentionally turned into a regular one. The rest of the scenarios are considered negative and would generate loss of time because of nurse errors made in ordering tests. (N+/P-) is a situation of over-requesting where time is lost for unnecessary tests; while without TNO, the patient would have been discharged right after the first consultation.(N+/P++) is a particular situation of under-requesting where triage nurse orders some necessary tests while missing others (the physician may order radiographs of other parts or different views of the same), which imposes additional trips to radiology. (N+/P-+) is a combination of both under and over requesting in which the patient is first sent unnecessarily to radiology by triage nurse, and then sent back later by the physician. In addition to potential loss of time, these situations have an impact on patient's convenience and satisfaction.

FIGURE 3 :

 3 FIGURE 3: POSSIBLE PATIENT PATHWAYS UNDER TNO

FIGURE 4 :FIGURE 5 :

 45 FIGURE 4: LOS IMPROVEMENT WITH TNO FOR HIGH REQUESTING RATES SCENARIOS

FIGURE 6 :

 6 FIGURE 6: SENSITIVITY ANALYSIS ON THE 4 KEY PROBABILITIES

FIGURE 7 :

 7 FIGURE 7: TNO EFFECTIVENESS DEPENDING ON ARRIVALS

FIGURE 8 :

 8 FIGURE 8: TNO EFFECTIVENESS DEPENDING ON ARRIVALS We derive the following insights:

TABLE I .

 I TNO ABILITY STATISTICS REPORTED IN THE LITERATURE

		[18]	[26]	[27]	Others
	Sample size	934	579	915
	on which
	TNO was
	applied
	Triage nurse	94.5%	72%	78%
	Requesting		(416/579)
	rate
	Over-	5.4% of	6.5% of	4% of	4.5% and 8%
	requesting	ordered tests	ordered tests	ordered tests	in [25] and
	rate (N+/P-)	considered	considered	considered	[21]
		unnecessary	unnecessary	unnecessary	respectively
		by physician	by physician	by physician
	Under-	65% of	47.2% of	23.5%
	requesting	situations of	situations of	(66/281) of
	rate (N-/P+)	no tests	no tests	situations of
		ordered by	ordered by	no tests
		nurse were	nurse were	ordered by
		followed by a	followed by	nurse were
		physician	a physician	followed by
		order (3.5% of	order	a physician
		total sample)	(13.3%	order (8% of
			of total	total)
			sample)
	Rate of	11% of already	5.3%	7.2% of	7.8% in
	additional	x-rayed	(22/416) of	already x-
	tests	patients	already x-	rayed
	requested by		rayed	patients
	physician		patients
	(N+/P++)

TABLE II .

 II THE POSSIBILITY TO APPLY TNO FOR THE MAIN TYPES OF TESTS

For which types of tests could TNO be

		relevant?
	X-ray	97%
	CT-scan	3%
	MRI	0%
	Echo	0%
	Blood tests	47%
	Urine analysis	83%
	ECG	83%