Marcos Dias De Assuncao

Carlos H Cardonha

Marco A S Netto

Renato L F Cunha

Marcos Dias De Assunc ¸ão

Impact of User Patience on Auto-Scaling Resource Capacity for Cloud Services

Keywords: Cloud computing, auto-scaling, resource management, scheduling

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Introduction

Cloud computing has become a popular model for hosting enterprise and backend systems that provide services to end-users via Web browsers and mobile devices [START_REF] Armbrust | Above the clouds: A Berkeley view of Cloud computing[END_REF][START_REF] Barga | The client and the cloud: Democratizing research computing[END_REF]. In this context, a typical scenario often comprises a provider of computing and storage infrastructure, generally termed as Infrastructure as a Service (IaaS) or simply a cloud provider; a service provider who offers a web-based service that can be deployed on Virtual Machines (VMs) hosted on the cloud; and the clients, or end-users, of such a service.

In this work, we investigate challenges faced by the service providers who compose their resource pools by allocating machines from cloud providers. Meeting end-users expectations is crucial for the service provider's business, and in the context of Web applications, these expectations typically refer to short requests response times. Simultaneously, there are costs associated with resource allocation, so resource pools with low utilisation are economically undesired. Moreover, clients demands are uncertain and fluctuate over time, so the problem of resource allocation faced by service providers is clearly non-trivial.

Elasticity, a selling point of cloud solutions, enables service providers to modify the size of resource pools in near real-time via auto-scaling strategies, allowing hence for reactions to fluctuations on clients' demand. Current strategies typically monitor and predict values of target system metrics, such as response time and utilisation level of relevant resources (e.g., CPU, memory, and network bandwidth), and employ rule-based systems to trigger auto-scaling operations whenever predefined thresholds are violated. The parameters employed by these strategies do not allow them to explore heterogeneity of expected response times and of tolerance to delays that end-users have towards service providers. In combination with actual response times, these two elements define the perception end-users have from a service QoS, which we will refer to as user patience.

Previous work in other domains has shown that endusers can present heterogeneous patience levels depending on their context [START_REF] Ramsay | A psychological investigation of long retrieval times on the world wide web[END_REF]. Moreover, in a society where human attention is increasingly becoming scarce, and where users perform multiple concurrent activities [4] 1and use multiple devices [START_REF] Venturini | Five insights into consumers' online video viewing and buying habits[END_REF] 2 , response time might not be the sole element defining the perceptions end-users have from a service's QoS. Our previous work investigated how application instrumentation [START_REF] Cardonha | Patience-aware scheduling for cloud services: Freeing users from the chains of boredom[END_REF][START_REF] Netto | Leveraging attention scarcity to improve the overall user experience of cloud services[END_REF][START_REF] Cunha | Exploiting user patience for scaling resource capacity in cloud services[END_REF], and end-user context and profiling [START_REF] De Assuncao | Context-aware job scheduling for cloud computing environments[END_REF] could be used in the The present work investigates how patience can be explored by auto-scaling strategies. Compared to our previous work [START_REF] Cunha | Exploiting user patience for scaling resource capacity in cloud services[END_REF], in addition to considering provisioning time, this article makes the following contributions:

• A scheduling algorithm and a patience-based autoscaling triggering strategy for a service provider to minimise the number of resources allocated from a cloud provider, applicable to scenarios where the maximum number of available resources is unbounded ( § 3);

• Experiments with bounded and unbounded numbers of resources available to service providers that show reductions of up to nearly 9% on resourcehour consumption compared to traditional autoscaling strategies ( § 4).

Problem Description

In this section we describe the scenarios investigated in this work, present the assumptions regarding capacity limitations on the pool of resources offered by the cloud provider, and explain the elements of client behaviour which are taken into account by our algorithms.

Figure 1 depicts the service hosting model and the main steps of auto-scaling operations. A service provider allocates a pool of VMs from a cloud provider to deploy its service, which is then accessed by endusers. The service provider and/or cloud provider periodically monitors the status of the pool, thus computing metrics such as resource utilisation. A metric lying outside certain lower and upper limits may trigger a scale in/out operation. A step size -the number of VMs to be allocated or released -is computed before a change to the capacity of the resource pool is requested.

Resources are pairwise indistinguishable, that is, they are VMs which have the same cost and performance characteristics. We also consider provisioning time; namely, after triggering the activation of resources, a service provider must wait for non-negligible time to use them. Finally, the scenarios either have a maximum number of resources that can be allocated (bounded) or do not have limitations of this nature (unbounded); these two scenarios are considered because they pose different levels of stress on resource utilisation.

We take into account that resources are paid only for the periods in which they were allocated in a per-minute billing model. This assumption is not unrealistic, as providers such as Microsoft Azure offer this type of service. Consequently, the strategies proposed in this article may allocate and deallocate a given resource several times within one hour in order to improve utilisation. This work also considers short tasks whose execution times are all equal and in the order of a few seconds, reflecting hence scenarios typically faced by providers of web services. Nevertheless, the concepts and results can, without any loss of generality, be reproduced in scenarios with either shorter or longer tasks. We assume that expectations on response time may vary across individuals, a realistic assumption in certain practical settings; for example, whereas end-users expect to get results from Web searches in a couple of seconds at maximum, clients performing large-scale graph mining operations, which can take from minutes to hours, tend to be more tolerant.

Impact on patience

QoS

Differences between expected and actual response times for submitted requests have an impact on users' patience (either positive or negative) whose weight decreases over time. More precisely, changes on the patience level of an end-user take place after the execution of each request and are described by a function applied to the ratio between expected and actual response times. Based on Prospect Theory [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF], we assume that the negative impact of delays on users' patience exceeds the benefits of fast responses (see Figure 2).

The service provider periodically evaluates the system utilisation and/or user patience and decides on whether the capacity of its resource pool should be expanded (shrunk) by requesting (releasing) resources from (to) the cloud. The questions we therefore seek to address are: (i) how to determine critical times when auto-scaling is necessary or avoidable by exploiting information on users' patience; (ii) how to determine the number of resources to be allocated or released; and (iii) how strategies behave under scenarios with and without restrictions on the maximum number of resources that can be allocated.

Algorithms for Auto-Scaling and Scheduling

This section starts with a mathematical description of the challenges a service provider faces to set its resource pool capacity. Based on this formalisation, we introduce the main contributions of this work: auto-scaling strategies and a scheduling algorithm based on user patience. Table 1 summarises the employed notation.

Mathematical Description

We consider a discrete-time state-space model where time-related values are integers in T ⊆ N representing seconds. There is a set U of end-users or clients and a sequence J of incoming requests to a service; the enduser who submitted request j ∈ J is denoted by u( j).

The service provider allocates up to m resources from the cloud provider in order to host its service and handle end-users' requests, and its resource pool capacity at time t ∈ T is denoted by m t for each t ∈ T ; recall that, with auto-scaling, m t may be different from m t for t t . We consider that m t ≥ b, where b ∈ N is defined a priori by the service provider; setting such a lower bound is a common practice in the industry to prevent resource shortage under low, yet bursty load. Moreover, w t represents the number of resources in use to handle requests at time t; note that w t < m t indicates that m tw t active resources are not being used by the service provider.

A request occupies one resource for ρ seconds in order to be processed, ρ ∈ N, and that request decomposition (i.e., execution of sub-parts in different machines or times) and preemption are forbidden. Due to the amount of time spent with queueing and scheduling operations, though, response times are almost always higher than ρ. Based on previous interactions with the service provider, the response time that end-user u expects for each submitted request is given by ρ × β u , where β u ∈ R is a multiplicative factor larger or equal than 1; we assume that β u does not change over time. The actual response time for request j is denoted by r j .

End-user u's patience level at instant t is denoted by φ t,u ∈ R + . If φ t,u is below a certain threshold τ u ∈ [0, 1), we say that u is unhappy. Based on the ideas introduced by Prospect Theory -where the sense of losing an opportunity transcends the feeling of gaining it -φ t,u behaves as follows. First, for every request j, u( j)'s reaction to response time r j is given by

x j = ρ × β u( j)
r j , where x j > 1 indicates that r j was smaller than expected, thus making u( j) positively "surprised"; x j < 1 accounts for cases where the r j was greater, making u( j) "disappointed"; and x j = 1 when equality between expected and actual response times hold. Considering the different user reactions suggested by Prospect Theory, we assume in this work that patience level φ t,u changes according to x j as follows:

φ t,u =        α 1,u φ t-1,u + (1 -α 1,u )x j , if x j > 1; α 2,u φ t-1,u + (1 -α 2,u )x j , otherwise,
where 0 ≤ α 1,u , α 2,u ≤ 1. Exponential smoothing is used in both cases to update the patience level. In order to make the immediate impact of slower response times higher than that of faster responses, we have α 1,u > α 2,u ; moreover, different end-users may have different values of α 1,u and α 2,u . In summary, each end-user u is described by a tuple (β u , α 1,u , α 2,u , τ u , φ 0,u ).

Auto-Scaling and Scheduling Strategies

In order to set an auto-scaling strategy, a service provider needs to define when an scaling operation should take place and to select adequate step sizes, which are the number of resources by which the resource pool capacity should extend or shrink.

Typically, resource utilisation is used to trigger scalein and scale-out operations. Resource utilisation at time t, denoted by υ t , is the ratio between the number w t of resource hours effectively used to handle requests and the number m t of resource hours allocated from the cloud (i.e., idle resources are not considered). The service provider sets parameters H and L, 0 ≤ L ≤ H ≤ 1, indicating utilisation thresholds according to which resources are activated and deactivated, respectively. Observe that, irrespective of scale-in decisions, though, resource pool capacity will never be inferior to b.

Ideally, resource pool capacity is modified before utilisation reaches undesired levels. For this reason, service providers typically rely on predictions of resource utilisation. In this work, this estimation is based on the measurements performed over the past i measurement intervals for some i ∈ N. Namely, after measuring υ t at time t, weighted exponential smoothing is used to predict the utilisation for step t + 1. If the past v ≤ i measurements (i.e., υ t-v , υ t-v+1 , . . . , υ t ) and the forecast utilisation are below (above) the lower (upper) threshold L (H), the scheduler triggers a scale-in (scale-out) operation.

We propose the following auto-scaling strategies:

• Utilisation Trigger (UT): a strategy that uses system utilisation and employs a formula to identify an upper bound on step sizes for activating/releasing resource capacity; • Unbounded-resource-pool Utilisation Trigger (UUT): uses system utilisation and employs a formula to determine an upper bound and a lower bound on step sizes for activating/releasing resource pool capacity, based on previous work [START_REF] Netto | Evaluating auto-scaling strategies for cloud computing environments[END_REF];

• Utilisation and Patience Trigger (UPT): extends the first strategy by using user patience to compute dynamically a factor for multiplying the step sizes suggested by the formula;

• Unbounded-resource-pool Utilisation and Patience Trigger (UUPT): extends the second strategy by using user patience in order to choose dynamically a value from the step size interval defined by the formula.

The rest of this section details the main principles of the auto-scaling strategies and the scheduling algorithm.

Utilisation-based Auto-Scaling

Let λ ∈ [0, 1] be the percentage of m by which the resource pool extends or shrinks in each auto-scaling operation, i.e., let s = λ × m . When setting utilisation thresholds L and H and step size s, service providers want to prevent the occurrence of opposite auto-scaling operations in consecutive steps. Such events take place in two situations, which can be avoided if we assume that w t = w t+1 as we discuss below.

First, we have a scale-out operation taking place at time t and a scale-in operation taking place at time t + 1 if w t m t > H =⇒ w t > Hm t , w t+1 m t +λ×m < L =⇒ w t+1 < L × (m t + λ × m), and w t ≤ w t+1 hold simultaneously. This can be avoided if the following inequality holds:

Hm t > L × (m t + λ × m) =⇒ H L > m t + λ × m m t =⇒ H L > 1 + λ × m m t .
A similar analysis indicates how to avoid scale-in operations at time t and scale-out operations at t + 1 for any t ∈ T whenever w t ≥ w t+1 . Namely, we want to avoid

w t m t < L =⇒ w t < Lm t and w t m t -λ×m > H =⇒ w t > H × (m t -λ × m) holding simultaneously; for this, we need H L > 1 + λ × m m t -λ × m . Moreover, we have that λ × m m t -λ × m > λ × m m t =⇒ m t > m t -λ × m =⇒ λ × m > 0,
so both inequalities are satisfied if w t = w t+1 and H L ≥ 1 + λ×m m t -λ×m . As the service provider will keep at least b machines active, we have that m t -λ×m ≥ b. Therefore, the parameters have to obey the relation below:

H L > 1 + λ × m b .
From the above, one derives an upper bound for the step size s = λ×m. This value is derived from borderline scenarios, though, so it yields step sizes that are too large in most cases. Therefore, service providers can choose a multiplicative factor γ ∈ [0, 1] indicating the strategy's "aggressiveness"; namely, given γ, the step size will be s × (1 -γ) and s × γ for scale-in and scale-out operations, respectively. Utilisation Trigger (UT) is an implementation of the ideas discussed above; namely, in order to use UT, a service provider only needs to set utilisation thresholds L and H and factor γ. Observe that this strategy is based solely on utilisation and assumes that m is finite.

When a service provider is not subject to upper bounds on the number of resources it can allocate (i.e., m = ∞), the formulae employed to derived upper bounds on step sizes used by UT cannot be applied. Under these circumstances, we employ Unboundedresource-pool Utilisation Trigger (UUT), a strategy introduced in our previous work [START_REF] Netto | Evaluating auto-scaling strategies for cloud computing environments[END_REF] in which the upper bound on s for scale-out operations is given by s ≤ m t υ t -L L and the lower bound by

s ≥ m t υ t -U U ,
whereas, for scale-in operations, the upper bound is

s ≤ m t L -υ t L and the lower bound is s ≥ m t U -υ t U .
Let L t and U t be the lower and upper bounds derived from the inequalities above at time t, respectively. In addition to L t and U t , UUT is also given an aggressiveness factor γ ∈ [0, 1]. Given these parameters, for scalein operations the value of s t is

s t = (1 -γ) × L t + γ × U t ,
whereas for scale-out operations it is

s t = γ × L t + (1 -γ) × U t .
More details are provided in previous work [START_REF] Netto | Evaluating auto-scaling strategies for cloud computing environments[END_REF].

Patience-based Scheduling and Auto-Scaling

Lowest Patience First (LPF) is a scheduling algorithm that orders arriving requests based on end-users' patience level as follows. If, at instant t, end-user u submits request j and x j < 1, we have

φ t+1,u = α 1 φ t,u + (1 -α 1 )x j ≥ α 1 φ t,u . Since lim r j →∞ ρ × β u r j = 0,
φ t+1,u becomes closer to α 1 φ t,u as r j → ∞. Therefore α 1 φ t,u is a lower bound for φ t+1,u , and the accuracy of this approximation grows as system load (and, consequently, response times) gets larger. Using this bound, we can estimate the minimum value k j ∈ N for which φ t+k j ,u becomes smaller than τ u as follows:

α k j 1 φ t,u ≥ τ u =⇒ α k j 1 ≥ τ u φ t,u =⇒ k j ≥ log α 1 τ u φ t,u ;
namely, value k j indicates the number of submissions for which u( j) will tolerate a bad QoS before having τ u( j) surpassed. LPF sorts the requests in the scheduling queue according to k j (smaller values first). In order to avoid starvation, k j is set to zero if request j is waiting for a period of time that is at least twice as large as the current average response time.

Under UT and UUT γ is fixed, so the service provider is forced to have the same behaviour (aggressive or conservative) in each auto-scaling operation. UPT and UUPT leverage UT and UUT, respectively, by employing user patience to set dynamically the value of γ.

Let Kt denote the average value of k j for all the requests in the service provider (either enqueued or in execution) at time-step t. Whenever the system decides that an auto-scaling operation should be performed, a sequence S containing the last k averages Kt is created and has some percentage of its largest and smallest values removed. From the remaining elements of S , the system takes the largest value K. Finally, the value of γ will be 1 -Kt K and Kt K for scale-out and scale-in operations, respectively.

Essentially, UPT and UUPT adapt the system behaviour according to users' current average-patience level. Namely, if Kt is high, a more aggressive policy is acceptable. Conversely, if Kt is low, a conservative approach could support the improvement of QoS.

Evaluation

We conducted extensive computational experiments in order to investigate the benefits and drawbacks of the algorithms presented in Section 3. We modelled scenarios where the maximum number of resources can be either bounded or unbounded, and evaluated the following combinations of task scheduling and auto-scaling triggering strategies:

• FIFO+UT or FIFO+UUT: First-In, First-Out (FIFO) scheduling with auto-scaling considering only resource utilisation.

• LPF+UT or LPF+UUT: LPF scheduling with auto-scaling considering only resource utilisation.

• LPF+UPT or LPF+UUPT: LPF scheduling with auto-scaling considering resource utilisation, and end-users' patience to define the resource step size of scale-out and scale-in operations.

Our comparisons employed the following metrics:

• Percentage of dissatisfactions: percentage of requests j whose response time r j led or kept enduser u( j)'s patience level below threshold τ u( j) ;

• Percentage of QoS violations: percentage of requests whose response times were longer than the amount of time expected by their end-users;

• Allocated resource-time: resource capacityin machine-seconds-allocated to serve user requests.

• Aggregate request slowdown: consists of the accumulated sum of all delays suffered by requests involved in QoS violations as is therefore given by j∈J max(r j -β u( j) × ρ, 0).

The rest of this section describes the experimental setup and discusses the obtained results.

Experimental Setup

A built-in-house discrete-event simulator was used to evaluate the performance of the proposed strategies. We crafted two types of workloads with variable hourlyload over a 24-hour period. The rationale behind these workloads, depicted in Figure 3, is as follows: • Normal day: consists of small peaks of utilisation during the start, middle, and towards the end of working hours. Outside these intervals, but still during working hours, the workload remains around the peak values, whereas outside the working hours the load decreases significantly.

• Peaky day: consists of tipping workload peaks, a scenario in corporations where requests need to be executed only at a few moments of the day.

The workloads are employed to determine the number of users accessing a service at a given time t. Every 10 seconds, the number a t of end-users accessing the service is adjusted to

a t = |U| × (l t + η),
where |U| denotes the maximum number of users who can be accessing the service at any given moment, 0 ≥ l t ≤ 1 is the load at time t according to the crafted workloads (representing hence a fraction of |U|), and η is uniformly distributed over -0.05 to 0.05 in order to add a random variation of 5%; the sum l t + η is always set to the closest value in the interval [0,1]. For instance, if at time t, |U| = 100, l t = 0.7 and η = 0.05, the number of users accessing the system and making requests is a t = 75. For intermediate values of t, a t = a t-1 . Unless otherwise noted, the experiments described here use |U| = 1, 000.

Requests have processing time ρ of 10 seconds, whereas end-user's expected response time for request j is ρ multiplied by β u( j) ≥ 1; e.g., if β u( j) = 1.3, the expected response time is 13 seconds. Request inter-arrival times, or end-users "think time"-amount of time in seconds separating the arrival of a response for request j and the submission of a new request by active end-user u( j)-is drawn uniformly from 0 to 100 for each request submission, that is, it is not constant for any end-user. The values of α 1 and α 2 are given by 0.2 and 0.1 multiplied by numbers drawn uniformly for each end-user from [0.9, 1.1], respectively. Finally, each τ u and φ u,0 are random numbers drawn uniformly from [0.45, 0.55] and [0.8, 1.0], respectively. These values are used by the algorithms employed by the service provider, so it can be assumed that they have been obtained or estimated based on historical data associated with each end-user.

We tested bounded scenarios in which maximum resource pool capacity was an even value between 162 to 180; in these cases, the initial capacity m 0 and the minimum capacity b are set to 25% of the maximum. For unbounded scenarios, initial/minimum capacity is set to 45 resources. The rationale behind these values is simple. Whereas it is a common practice in the industry to set a minimum resource capacity for a service, we do not want this minimum capacity to be large enough to eclipse the impact of auto-scaling decisions.

The lower and upper target thresholds for resource utilisation (i.e., L and H) are 40% and 70% respectively, which are default utilisation levels used by existing auto-scaling services, such as Amazon Elastic Compute Cloud (EC2). Resource utilisation levels are measured every 60 seconds. The value of γ is set to 0.25 for strategies UT and UUT under bounded scenarios and to 0.5 under unbounded scenarios, respectively.

Provisioning time in each scale-out operation is measured in seconds and is drawn from a Normal distribution with mean 180 and standard deviation 15; sampling is performed in an aggregate way, that is, a single value is used by all resources involved in the same operation. These values were chosen based on existing work that identified that the mean time required to provision lean VM instances in certain clouds is often over 100 seconds [START_REF] Mao | A performance study on the VM startup time in the cloud[END_REF]. Moreover, the fact that in a group of VMs there are certain VMs that can boot slower and that the provision of VMs with Web server software stack takes longer as demonstrated in the existing work, we added extra 80 seconds. We performed experiments with higher mean values (up to 10 minutes) and obtained similar conclusions for the proposed techniques (therefore, due to space constraints, we report only on results involving provisioning time variable to 180±15). Deallocation time was not considered in this work because it is typically much shorter than provisioning time and its effects can only be observed in scenarios where utilisation changes abruptly and quickly, which is not the case of the workloads considered in this work.

Finally, we simulated the equivalent of 4 days of activities of a service provider in order to reduce biases that single executions could have brought to the final results.

Result Analysis

Bounded number of resources. We start discussing the results of our experiments with scenarios where the maximum resource pool capacity is bounded. These results are summarised in Figures 4,5 Figures 4 and 5 present the percentage of requests whose response times were larger than expected. In general, LPF+UPT is superior and to FIFO+UT in scenarios with 162-170 and worse than FIFO+UT with 172-180 resources, respectively, whereas LPF+UT is better than both in all scenarios with respect to this metric. Poor results from FIFO+UT are not surprising, as it is completely oblivious to user patience. The comparison between LPF+UT and LPF+UPT is less straightforward; LPF+UT leads to fewer QoS violations than LPF+UPT, but Figures 6 and7 show that the later compensates this with significant reductions on allocated resource time (approximately 9%).

Additionally, results of LPF+UPT on percentage of QoS violations and on percentage of dissatisfactions for scenarios with 170-180 resources show the effects of making γ dynamic. Namely, in scenarios where the maximum number of resources is large enough to satisfy demand, users' patience level are more likely to stay high. Whenever auto-scaling operations have to be performed, the algorithm assigns "aggressive" values to γ that enforce the allocation (deallocation) of a very small (large) number of machines in scale-out (scale-in) operations. As a result, response time for a certain number of requests may be larger than expected in situations where such delays could have been avoided. However, the results presented in Figures 8 and9 for 172-180 resources suggest that it is possible to provide insufficient QoS for approximately 1-2% of all requests without perceptible damages to user patience in such scenarios (as illustrated by LPF+UPT in Figures 4 and5).

Figures 8 and9 show a clear superiority (i.e., lower percentages) of LPF+UT and a clear inferiority of FIFO+UT with respect to end-user dissatisfaction. The latter phenomenon can be again explained by the fact that FIFO does not take patience into account. Moreover, differences between LPF+UT and LPF+UPT are again counterbalanced by differences on the volume of resource-time usage for each configuration. The results of scenarios with 174-180 resources are inconclusive, as the percentage of dissatisfied users in these cases is insignificant; because queues are almost never formed in these situations, the scheduling strategies basically deliver the same solutions and auto-scaling algorithms have enough time to adapt the resources' pool.

Finally, the results show that keeping user patience in satisfactory levels is more challenging for the Peaky workload. The overall resources demand on both scenarios are different, but whereas differences on the volume of allocated resources are relatively low (approximately 10%), the percentages of QoS violations and dissatisfactions differ by a factor of 2. Peaky workloads model situations where resource demand increases abruptly, and under these circumstances, the number of QoS violations tend do increase considerably. Therefore, it is natural to expect lower user satisfaction levels in such scenarios. Unbounded number of resources. We present now results involving the simulation of scenarios where service providers could provision as many resources as they wished. These results are summarised in As expected, the number of QoS violations is considerably smaller in these scenarios. Figure 10 shows that the occurrence of such events is extremely rare in both workloads (< 1%). One can also observe a potentially statistically negligible, but nevertheless interesting, phenomenon in this figure: all the strategies yielded more QoS violations in the Normal workload than in Peaky. Moreover, differences on the percentage of dissatisfaction presented in Figure 12 reinforces this apparent contradiction. The explanation for these results lay in an observation that was irrelevant in scenarios where violations take place frequently: although its changes are more abrupt, Peaky is smoother, in the sense that modifications on its upwards/downward orientation are less frequent than in Normal (6 against 15); QoS vi-olations are more likely to occur during these shifts, which justifies the observed difference. Additionally, Figure 10 shows that FIFO+UT delivered less QoS violations than the other two strategies for the Peaky workload; this happened because patience-aware strategies focus on the minimization of user dissatisfaction rather than QoS violations, and sacrificing the latter may be necessary in order to excel on the first. Conversely, results presented in Figure 12 suggest that the impact of QoS violations on end-user patience on Normal was considerably larger than on Peaky. Direct comparison with Figure 10 suggests that the number of violations play a major role on this phenomenon, so we investigated the aggregate request slowdown.

The results are presented in Figure 13 and show that the values for Peaky and Normal are close; consequently, individual QoS violations typically involved much larger delays on Peaky than on Normal. From this, we conclude that having several small delays is more harmful to user dissatisfaction than a few long delays. Finally, Figure 11 reinforces the benefit of using a patience-aware auto-scaling triggering strategy. Whereas user dissatisfaction levels for all configurations are basically the same in both workloads, LPF+UPT consumed 9% less resources than the other strategies.

To evaluate the impact of provisioning time on the auto-scaling strategies described in this work, we performed an experiment with provisioning time of 10 ± 5 seconds and 10,000 as a maximum number of end-users -that is, |U| = 10, 000 when computing the number of users in the system a t at time t. The minimum number of resources, b, was set to 450. This scenario presents demand peaks that exceed 2,000 VMs. Figure 14 and Figure 15 summarise the results for allocated-resource time and percentage of dissatisfactions, respectively.

We can observe that the patterns identified in previous experiments hold under this last scenario; there is a reduction in the amount of allocated resource time resulting into a small increase in the number of requests that suffer from insufficient QoS. Further investigation revealed that this is the case because, although resources can be provisioned quickly, LPF delays autoscaling operations by exploiting users patience levels, eventually leading to dissatisfactions as resource utilisation and triggering of auto-scaling decisions are taken at discrete intervals. Although the patience-based autoscaling strategy could be made more responsive to variations in resource utilisation, that would lead to an increase in the cost with infrastructure as resources would be activated and released more often, and eventually render the consideration of users' patience levels in auto-scaling unusable.

Related Work

Projects related to our work fall into categories of scheduling, user behaviour, and cloud computing autoscaling. Scheduling is a well-studied topic in several domains for which a large number of theoretical problems, solutions, heuristics, and practical applications have been considered [START_REF] Blazewicz | [END_REF][START_REF] Pinedo | Planning and Scheduling in Manufacturing and Services[END_REF]. Some of the used algorithms include FIFO, priority-based, deadline-driven, hybrid approaches that use backfilling techniques [START_REF] Tsafrir | Backfilling using system-generated predictions rather than user runtime estimates[END_REF], among others [START_REF] Braun | A comparison of eleven static heuristics for mapping a class of independent tasks onto heterogeneous distributed computing systems[END_REF][START_REF] Feitelson | Theory and practice in parallel job scheduling[END_REF]. In addition to priorities and deadlines, other factors have been considered, such as fairness [START_REF] Doulamis | Fair scheduling algorithms in grids[END_REF], energy-consumption [START_REF] Pineau | Energyaware scheduling of bag-of-tasks applications on master-worker platforms[END_REF], and contextawareness [START_REF] De Assuncao | Context-aware job scheduling for cloud computing environments[END_REF]. Moreover, functions were used to model how the importance of results to users varies over time [START_REF] Lee | Precise and realistic utility functions for user-centric performance analysis of schedulers[END_REF][START_REF] Auyoung | Service contracts and aggregate utility functions[END_REF].

User behaviour has been explored for optimising re-source management in the context of Web caching and page pre-fetching [START_REF] Galletta | Web site delays: How tolerant are users?[END_REF][START_REF] Alt | Bridging waiting times on web pages[END_REF][START_REF] Atterer | Knowing the user's every move: user activity tracking for website usability evaluation and implicit interaction[END_REF][START_REF] Cunha | Determining www user's next access and its application to pre-fetching[END_REF][START_REF] Bhatti | Integrating user-perceived quality into web server design[END_REF][START_REF] Bouch | Quality is in the eye of the beholder: meeting users' requirements for internet quality of service[END_REF]. The goal in previous work was to understand how users access Web pages, investigate how tolerant users are to delays, and pre-fetch or modify page content to enhance user experience. Techniques in this area focus mostly on adapting Web content and minimising response time of user requests. Service research has also investigated the impact of delays in users' behaviour. For instance, Taylor described the concept of delays and surveyed passengers affected by delayed flights in order to test various hypotheses [START_REF] Taylor | Waiting for service: the relationship between delays and evaluations of service[END_REF]. Brown et al. and Gans et al. investigated the impact of service delays in call centres [START_REF] Brown | Statistical analysis of a telephone call center: A queueing-science perspective[END_REF][START_REF] Gans | Telephone call centers: Tutorial, review, and research prospects[END_REF]. In behavioural economics, Kahneman and Tversky [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF] introduced Prospect Theory to model how people make choices in situations that involve risk or uncertainty. Netto et al. introduced a scheduling strategy that considers information on how quickly users consume results generated by service providers [START_REF] Netto | Leveraging attention scarcity to improve the overall user experience of cloud services[END_REF]. Our previous work investigated the scheduling of user requests considering their patience and expectations, but with no auto-scaling of cloud resources [START_REF] Cardonha | Patience-aware scheduling for cloud services: Freeing users from the chains of boredom[END_REF]. Shen et al. introduced a system for automating elastic resource scaling for cloud computing [START_REF] Shen | Cloudscale: elastic resource scaling for multi-tenant cloud systems[END_REF]. Their system does not require prior knowledge on the applications running in the cloud. Other projects consider auto-scaling in different scenarios and proposed several approaches, such as modifying the number of resources allocated for running MapReduce applications [START_REF] Chen | Interactive analytical processing in big data systems: A crossindustry study of mapreduce workloads[END_REF][START_REF] Fadika | DELMA: Dynamically ELastic MapReduce Framework for CPU-Intensive Applications[END_REF], comparing vertical versus horizontal auto-scaling [START_REF] Sedaghat | A virtual machine re-packing approach to the horizontal vs. vertical elasticity trade-off for cloud autoscaling[END_REF], minimising operational costs [START_REF] Mao | Auto-scaling to minimize cost and meet application deadlines in cloud workflows[END_REF], and providing integer model based auto-scaling [START_REF] Mao | Cloud autoscaling with deadline and budget constraints[END_REF] .

Unlike previous work, our proposed auto-scaling technique considers information on user patience while interacting with a service offered by a service provider using resources from a cloud infrastructure.

Conclusions and Future Work

In this article, we extended our investigation on patience-based algorithms for auto-scaling strategies. Traditional mechanisms are based solely on resource utilisation information and other system metrics, ignoring actual users' needs with respect to their desired QoS levels. As a consequence, resources are overprovisioned under scenarios where they are not strictly needed; that is, where additional resources do not lead to significant improvement in QoS.

We conducted extensive experiments using FIFO and Lowest Patience First as scheduling algorithms, and auto-scaling triggers that rely either on utilisation or user patience and utilisation. Our experiments considered scenarios where the maximum number of resources could be either bounded or unbounded, and different versions of the auto-scaling algorithms were designed to cope with both possibilities. Our experiments considered non-trivial (i.e., larger than 0) provisioning time.

Experimental results suggest that patience-based strategies can indeed provide significant economic gains to service providers. Namely, we observed that these strategies were able to reduce resource-time usage by approximately 9% while keeping QoS at the same level as those based solely on utilisation. We conclude that the identification of unnecessary buffers in QoS levels that will not necessarily improve user experience in cloud environments is a key element for service providers, as it may enable savings in infrastructure costs and bring competitive advantage for adopters.

We believe the proposed strategies fill an important gap in the literature. As sensors and Internet of Things become more pervasive, more data will be available that can be used to determine factors that influence users' patience. By using this data, one could certainly specialise and improve the techniques presented here. With respect to resource management, the strategies presented in this work handled horizontal resource elasticity and assumed that resources are pairwise indistinguishable. In future work, we aim to relax such assumptions, handling vertical elasticity and the allocation of multiple types of virtual machine instances.

Figure 1 :

 1 Figure 1: Service provision model and auto-scaling scenario considered in this work.

(

  Above expectation) (Below expectation)

Figure 2 :

 2 Figure 2: Prospect theory modelling variations of user patience as a function of QoS [10]; where QoS can represent expectations on, for instance, request response time and jitter.

Figure 3 :

 3 Figure 3: The workloads considered in this work, where the hourly load l is employed to determine the number of end-users accessing the service.

Figure 4 :Figure 5 :

 45 Figure 4: Bounded: Percentage of QoS violations for FIFO+UT, LPF+UT, LPF+UPT on the Normal workload.

Figure 6 :Figure 7 :

 67 Figure 6: Bounded: Comparison of resource-time usage for FIFO+UT, LPF+UT, LPF+UPT on the Normal workload.

Figure 8 :

 8 Figure 8: Bounded: Percentage of user dissatisfaction for FIFO+UT, LPF+UT, LPF+UPT on the Normal workload.

Figure 9 :

 9 Figure 9: Bounded: Percentage of user dissatisfaction for FIFO+UT, LPF+UT, LPF+UPT on the Peaky workload.

Figure 10 :

 10 Figure 10: Unbounded: Percentage of QoS violations for FIFO+UT, LPF+UT, LPF+UPT.

Figure 11 :Figure 12 :

 1112 Figure 11: Unbounded: Comparison of resource-time usage for FIFO+UT, LPF+UT, LPF+UPT.

Figure 13 :

 13 Figure 13: Unbounded: Aggregate request slowdown for FIFO+UT, LPF+UT, LPF+UPT.

Figure 14 :Figure 15 :

 1415 Figure 14: Unbounded: Comparison of resource-time usage for FIFO+UT, LPF+UT, LPF+UPT under provisioning time of 10±5 seconds and |U| = 10, 000 users.

Table 1 :

 1 Summary of the notation used in this article.

	Symbol	Description
	T	Discrete-time state-space model
	U	Set of end-users of the service
	J	Sequence of requests made by end-users
	u( j)	End-user who submitted request j
	m	Service provider's maximum resource pool capacity
	m t	Service provider's resource pool capacity at time t
	b	Service provider's minimum resource pool capacity, set a priori
	w t	Number of resources in use at time t
	ρ	Processing time of a request
	β s t	Resource step size at time t
	γ	It determines the aggressiveness of auto-scaling strategies when computing the step size s t
	k j	Number of responses with bad QoS u( j) will accept before having patience limit exceeded
	Kt	Average value of k j for all the requests in the service (either enqueued or in execution)
	τ u( j)	Maximum response time a user u is willing to endure after receiving a response for j
	a t	Number of end-users accessing the service at time t
	L t	Lower bound derived for step sizes in scenarios where m = ∞
	U t	Upper bound derived for step sizes in scenarios where m = ∞
	l t	System load at time t according to workloads

u Factor that determines the expected response time end-user u has after submitting requests r j Actual response time for request j φ t,u End-user u's patience level at time t x j End-user u( j)'s reaction to response time r j α 1,u and α 1,u Parameters that define how the patience level of end-user u varies over time υ t Resource utilisation at time t L Lower bound for resource utilisation H Upper bound for resource utilisation i Number of intervals for predicting resource utilisation λ Percentage of m which equals step size s Resource step size (i.e., number of resources to activate/deactivate in an auto-scaling operation)

Ofcom reports that up to 53% of UK adults media multi-task while watching TV.

Accenture's study shows users expect other devices to augment content shown on TV.

Acknowledgements

Some experiments presented in this paper were carried out using the Grid'5000 experimental testbed, being developed under the Inria ALADDIN development action with support from CNRS, RENATER and several Universities, as well as other funding bodies (see https://www.grid5000.fr). This work has also been supported and partially funded by FINEP / MCTI, under subcontract no. 03.14.0062.00.


Adrien Lutz

Non-Euclidean geometry and political economy How Jacques Rueff explained unemployment in England (1919-1931)

Keywords: Rueff, unemployment, real wage B13, E24, N00 B13, E24, N00 Rueff, unemployment, real wage

This paper provides new perspectives on the French liberal economist Jacques Rueff , especially as regards his early writings on unemployment. We aim to show that Rueff distinguishes the root causes of permanent unemployment in England (1919)(1920)(1921)(1922)(1923)(1924)(1925)(1926)(1927)(1928)(1929)(1930)(1931) based upon an interesting reading of non-Euclidean geometry. Controversially, this enables him to locate the cause of unemployment in the stickiness of the wage/price ratio. Hence, arguing that reality remains inaccessible in itself, Rueff focuses on a succession of variables (price, wage, unemployment), supplemented by his concepts of rational ego and the reasoning machine, in order to approach this reality.

Introduction

The works of French economist Jacques Rueff (1896Rueff ( -1978) ) are well known and their importance is widely acknowledged. As indicated by his preface to Robbins's La grande dépression (1935), many famous economists of Rueff's era respected his ideas, addressing for example his views on unemployment (Beveridge 1909, Hicks 1932, Keynes 1939[START_REF] Pigou | The Economic of Welfare, London: Mac Millan. -1927[END_REF][START_REF]Chômage et salaires réel en Angleterre dans la période 1919-1925, Séance de la Société d'Économie Sociale du 14 décembre 1925[END_REF], money [START_REF] Mundell | The monetary consequences of Jacques Rueff[END_REF] or transfers (Keynes 1929a, b, c). More recent works on Rueff have focussed on the monetary aspects of his writings [START_REF] Pays | Libérer la Monnaie: les Contributions Monétaires de Mises[END_REF], Chivvis 2010).

However, the young Rueff (pre-1939) remains largely unknown, and the use he proposed for non-Euclidean geometry in political economy has been little studied (excepting Frobert 2010). This is due to the fact that Rueff is influenced by a Cartesian method inherited from geometry to the problem of unemployment in England (Rueff 1925(Rueff , 1931)), which caused controversy -generally speaking, all that is recalled about this episode is that he made some proposals which seemed rather cold-blooded compared with the human disaster of unemployment. Although Rueff is respected by his peers, the anomaly remains that he is a major French economist whose early epistemological writings are very little known (see, however, Claassen 1967). Moreover, his interest in geometrical methods is widely neglected. This article remedies this neglect by providing an overview of Rueff's readings of geometry in political economy, and exploring its intellectual roots.

Rueff is strongly influenced by a metaphor based upon non-Euclidean geometry. More exactly, he adopts a particular conception of reality: only the structure of reasoning counts, regardless of whether the result is observable or not. Here he makes reference to the scientific developments in non-Euclidean geometry achieved by Carl Friedrich Gauss (1777-1855), Bernhard Riemann (1826-1866), János Bolyai (1802-1860) and Nikolai Ivanovich Lobachevsky . Jacques Rueff is not the only economist to have drawn on this movement to bring new perspectives to the science; but we argue here that Rueff is the only one to have been strongly influenced by this field of knowledge in order to enlighten social issues.

In this respect, the first part of this paper deals with the concept of unemployment. Here, Rueff provides a controversial analysis based upon the tools he designed, particularly the wage/price ratio. Indeed, the very way he constructed his notion of causality -based upon the difference between the level of unemployment and this famous ratio -was seen as controversial. Hence, a major consequence is that Rueff argues for a decrease in the wage/price ratio: here, then, stands a complex choice, decreasing wages while prices stay constant, or increasing prices while wages stay constant.

The second part of this paper shed light of the intellectual root of this analysis and states that the tools designed by Rueff are based upon his interest in non-Euclidean geometry, a development in mathematics which indicated that our world could be apprehended from more than three dimensions. This intellectual movement remains the first step taken by several sciences towards a new conception of reality. We also show that Rueff considers reality to be inaccessible. In his view, we only can design tools which tend to approach this reality. Hence, our rational ego and our reasoning machine -in addition to our languageenable us to build knowledge in a way which leads us to understand our human condition. Rueff never places trust in our ability to find principles of causality: causes are only a human creation, a convention which helps us in our practice of science2 .

Rueff's analysis of British unemployment

We analyse here Rueff's work as related to unemployment, published as "Les variations du chômage en Angleterre" (1925), "Chômage et salaire réel en Angleterre dans la période 1919-1925" (1926) and "L'assurance-chômage, cause du chômage permanent " (1931a, 1931b). 3 These works contain the clearest presentation of his method. We separate our development into two key syllogistic aspects, as Rueff himself recommended: observation (4.1) and reasoning (4.2).

The empirical branch: hypothesis and observations

Rueff's work starts from two simple observed facts: the almost exponential increase in unemployment in England between 1919 and[START_REF] Pigou | The Economic of Welfare, London: Mac Millan. -1927[END_REF] the parallelism between the unemployment curve and the wage/price ratio curve. Therefore, Rueff hypothesizes, changes in unemployment are linked to the difference between levels of wages and prices. More precisely, these changes in unemployment could be explained by changes in this difference and follow the same fluctuations. Before going further, we thus present the variables unemployment, price and wage:

1. Unemployment is a rate established in relation with the unions: hence it does not strictly measure the number of individuals over 15 years of age who are without a job and who are searching for one, as it is defined today. Rather, this rate is the "percentage of workers unemployed in a number of trade unions compared with the total number of members of these unions " (1931a: 245). This percentage was commonly used at that time and doesn't aspire to pose any particular concern as regards its design or in its use. Hence, "the published indices can't give an idea of the absolute importance of unemployment, but enable us to follow the magnitude of its variations in satisfactory condition " (1925b: 426).

2. Concerning wages, Rueff reasons with real wages, which are obtained by calculating the ratio between nominal wages and prices. For nominal wages, he uses an index created by Professor Arthur L. Bowley (1869Bowley ( -1957)), a statistician and a major figure in the introduction of descriptive statistics in political economy (few works of his time use numbers other than his own).

3. On prices, Rueff uses wholesale prices furnished by the Board of Trade: this is the monthly sum of the "price of a number of commodities and goods, chosen once and for all (45 in the index of Statistique Générale de France)" (1923a: 101). Again, this variable is commonly used and seems difficult to criticize.

The controversial element is the ratio itself. Indeed, Rueff has been criticized for using a real wage based upon wholesale prices: as Rueff argues, the goal is to analyse the "disparity between changes in wages and changes in selling prices, not the purchasing power of the worker […] without going into considerations of fairness or fair price " (1931c: 244). Hence, what really interests Rueff is the observation of the disparity between different variables. Rueff doesn't see wages as purchasing power but as a cost of business returns, and so as a "cost of human labour per unit of output" (Ibid., 244). Rueff focuses clearly on the supply side and on changes in wages as a cost as well as the selling prices. This explains the choice of wholesale prices. 3 See also the other works from Rueff on unemployment in which (with Georges Lane) he analyses English unemployment after the Second World War (from 1963 to 1975) and criticizes the Phillips's curve (see their article, "La fin de l'ère keynésienne", in Le Monde, February 19, 20 and 21 th 1976).

According to Rueff, the index of wholesale prices remains the most able to establish observations in terms of the economic equilibrium of the goods market. In fact, "we see the price index increasing with the overall average of prices, and we can say that when this index increases, the purchasing power of the monetary unit, which is used to express these studied prices, decreases in the same proportion" (1923a: 101). Indeed, this is a conception of price as a factor of economic equilibrium. A price is adequate even though it remains high, simply because it has been freely determined. In this respect, wage acts as a variable of equilibrium on the labour market. Rueff notes, therefore, several elements that constitute the empirical branch of this work and which are the basis for his reasoning (see Table 1):

It is precisely from the time when the index of wholesale prices decreases in England as unemployment rises, and it continues to increase as the wage index increases. Then the representing curve of this index is close to the prices curve, and unemployment begins to decrease, until the beginning of 1925 when it resumes its increase as the difference between the two curves grows again. (1925b: 427) The indexes of wages W and prices P (W/P as a real wage) both increase until the mid-1920s with the level of variation of the price index a third higher than the wage index, all with low unemployment. Then the prices index decreases more than wages. Moreover, the decrease of prices arises earlier compared to wages (mid-1920 against mid-1921) and the wage/price ratio increases sharply, as well as unemployment. Finally, the price index continues to decline, although it remains low, while the wage index stabilizes. Unemployment only decreases by some 8.8 points before resuming its rise, while the W/P ratio also arises again. The second quarter of 1920 sees unemployment reaching its minimum (1.1%) while the price index also reaches its maximum, where the wage is at its 5 th highest value (after the next 4 quarters). This period of growth of these two variables is symbolized by a W/P ratio at a low level (the 3 rd lowest value after the two previous quarters). Exactly one year later, the second quarter of 1922 is marked by one of the highest unemployment rates (20.9%). It is interesting to note that, over the period, the wage index reaches its second highest value (after the next quarter), while the price reaches its 18 th highest value, giving rise to the maximum ratio wage/price (1.333).

And this is easily explained. During a monetary depreciation, it is due to the constant rise in prices that the workers are demanding an increase in their wages, and they obtain it with an obvious but significant lateness. When, for example, the price level reported to base 100 for the year 1913 reaches its maximum, i.e. 324 (more than three times the pre-war level), the level of wages reported on the same basis never exceeds 250 (two and a half times the pre-war level). Here stands a general situation that seems characteristic of all periods of rising prices. (1925b: 435) The wage index begins to decline when unemployment reaches its maximum, while the price index begins to decrease when unemployment reaches its minimum, with unemployment ceasing to decrease from mid-1921 to mid-1924. This decrease in index continues -except for some minor variation -in the last two years when the wages decline, and where prices had already seen their rate of decline slowing significantly from 1922. This coincides with a rise in unemployment from mid-1924. Regarding the wage/price ratio, we note that it decreases while unemployment reaches its minimum. Then it increases along with it: this seems to be due to the increase in the wage index coupled with the decrease in prices until mid-1921, and then to a lesser decrease in the wage index related to prices until the beginning of 1922. Unemployment then fluctuates downward (-19.6%) from 1922 to 1923, and then increases (12.2%) from 1924 to 1925. This may be explained by the above observations, namely a small rise in the wage index in 1924 accompanied by a slowdown in the rate of decrease of the price index starting at the beginning of 1922.

Obviously these figures are not accurate and only approximately reflect general trends, but this does not prevent us seeing, for instance, the impact of the American crisis of 1920-1921: this is in part due to lower discount rates (bank rate) in the United States, intended to absorb part of the debt. The result is a decrease in prices and in economic activity that inevitably affects the English markets. Hence, these figures do not teach us much in themselves, except in so far as they are indicative of an a priori link between the wage/price ratio and unemployment. It would be naive to hope for a return to a low unemployment rate with a level of the price index a third higher than wages, as it was the case until the middle of 1920.

With his tools defined and the variations discussed, Rueff then derives a possible link between changes in unemployment and in the ratio between wages and prices. He then constructs a curve for this second ratio (see diagrams 1 and 2). What are his observations?

You just have to simultaneously consider the two curves to determine, without a doubt, the whole synchronism of their movements. In general the unemployment index varies along with the wage/price ratio except at some points (Q1 1920 Q1 1924) where changes in the first immediately follow the second. (Ibid.,429) In his second article of 1931, Rueff notes that "a simple glance at the diagram shows clearly that what was true from 1919 to 1925 was also true from 1928 to 1931, and with as much rigor and precision as in the past" (1931a: 214). In the end, Rueff correlates the wage/price ratio and the unemployment rate. This correlation was described by Sir Josiah Stamp (1880-1941) -statistician and former director of the Bank of England from 1928 to 1941 -as particularly significant (95%) in the Financial Times of March 15 th 1926. Therefore Rueff bases his hypothesis upon the observation of general facts.

The rational branch: Rueff's reasoning

The hypothesis is that unemployment increases as the difference between wages and prices also increases. As prices freely fluctuate -excluding some protectionist tendencies whose incidence has little to do with the effects of unemployment insurance -questions arise concerning sticky wages. Rueff thus wonders whether wages are too high, or more precisely why wages does not play their role by stabilizing an economic equilibrium.

There is, thus, a reason to believe that the immediate cause of widespread unemployment […] would constitute a failure of wage adjustment to the general price level. In so far as a crisis unprecedented in History rages in England, it is that lower prices are only followed by lower wages -and that after stabilization at the end of 1921, the percentage increase in wages relative to the pre-war remained too high relative to the percentage increase in price. (1925b: 432-433) The price mechanism, which guarantees the economic equilibrium, is blocked to the extent that labour supply is higher than demand. This occurs because of an imbalance in the labour market, where the variable-price is unable to address this deficit adjustment:

The discipline of the trade unions in the first place, is extremely powerful in England, and the contract system of collective work, more widespread than elsewhere. The tradition, however, would have been insufficient to maintain the resilience of unemployed workers to the inevitable movement of wages, if a policy of subsidies to the unemployed, as generous as costly to the country, had allowed them to remain unoccupied indefinitely, rather than infringe the union instructions. (Ibid.,(435)(436) So this is due first to the role played by the unions and also by the state, which act to impede the stabilizing role of the price mechanism. Here lies the core of Rueff's reasoning: for this observation leads ultimately to the postulation of different types of unemployment linked with sticky wages:

1. The context is characterized by a cyclical decrease in prices higher than wages, "hence we see how there can exist in the lowering or increasing of prices an influence able to govern the labour market" (Ibid., 435). Rueff here joins the classical tradition regarding the interdependence of labour, monetary, goods and financial markets. To this type of unemployment is added the movements of labour as "in the month of August 1920, there were 120,000 unemployed persons in Great Britain, a quite similar figure to the pre-war figure, which seemed to respond to the inevitable movement of labour" (1926: 54). These inevitable movements of labour, as well as the movements of prices, constitute what Rueff calls temporary unemployment. This is defined as unemployment due to lower prices being absorbed by an increase in activity through lower wages.

2. There is an excessive wage compared to any equilibrium value given by the performance of work. This constitutes a classic vision of voluntary unemployment. So there would be, on Rueff's account, inactivity traps, as "it is very curious and seemingly contrary to all economic laws, that wages could remain stable, while the labour supply also significantly exceeded labour demand " (1925b: 435). This type of unemployment is permanent in the light of previous developments: it persists since the variables of prices and wages do not freely fluctuate.

Finally, temporary unemployment isn't a problem in itself as, according to Rueff, the wages/prices ratio varies with unemployment itself. This allows a return to equilibrium: there is often unemployment in periods of decreasing prices. This unemployment is only temporary, as it is absorbed when prices rise on the condition that wages follow the trend. For Rueff, this has always been the case. But this wages/prices ratio is now blocked, or more precisely it doesn't vary appropriately: wages don't decrease quickly enough, thus establishing permanent unemployment. The problem here is not low wages:

From now, regarding wages the level wouldn't be one that would ensure the employment of all the available labour force -given the volumes of existing capital -but the resulting level from the rate of unemployment insurance, itself determined by social rather than economic considerations. Thus, in England the level of wages is found strictly immobilized. (1931c: 871) Here he is targeting the first type of unemployment noted above. In other words, the disappointments of British economic policy are due to the fact that the lawmakers are taking into consideration factors that are not purely economic. They are dealing with social considerations that are not relevant to the problems they are seeking to tackle. Political economy should only deal with economic facts, whereas the British government is confusing economic and social views. This is the heart of the problem, as it prevents market regulation of prices and so gives birth to inactivity traps. We should also notice that unemployment is not the only problem here, as "the Bank of England has consistently tried for two years to avoid the higher interest rates that would rather result from the deficit in the balance of payments" (Ibid., 873).

Rueff takes the opportunity in his article of 1931 to attack the work of the ILO (International Labour Office), an institution which he thinks has not seen this mismatch: its analysis takes the level of prices to be the sole cause of unemployment (see Annex II of Rueff 1931a: 247-251, entitled Notes relatives aux vérifications expérimentales présentées par le Bureau International du Travail dans son étude sur "Les fluctuations monétaires et le chômage"). In addition, in the ILO's analysis of the causes of unemployment, it fails to deal with what restrains wages -an omission noticed by Rueff, who remarks that this report "didn't mention, at least explicitly, measures which tend to immobilize wages" (1931a: 223). The ILO isn't entirely wrong in its analysis, as it is a fact that there is much unemployment as a result of lower prices (an equivalent to Rueff's temporary unemployment), but when prices increase, unemployment doesn't decrease, since the wages/prices ratio continues to increase, as Rueff has shown. Indeed, from mid-1923 wages increase simultaneously with a decrease in prices. Hence, according to Rueff, the ILO focuses too much on the variable of price and obscures the wage/price relationship. If wages are blocked, it is normal that unemployment should vary based upon price alone, but the fact remains that it is the lack of wage adjustment which is at issue, according to Rueff.

Regarding possible solutions, Rueff doesn't consider a decrease in the dole (unemployment insurance), but rather advocates higher prices without increasing wages or lower wages without a decrease of prices. In order to avoid an impact on the entire global market, a depreciation of the British Pound or an increase tariffs appear as good solutions. Rueff proposes this second solution in his first article (1925), but only six years later it is no longer conceivable for him. This is due in part to the protectionist inclinations of England, but is also based on the observation that prices have increased without a concomitant decrease in unemployment.

Hence Rueff have designed tools in order to enlighten social issues such as unemployment. We argue that this is based upon his early readings -for instance Poincaréand his interest in non-Euclidean geometry. Indeed, this field of knowledge was in progress during his youth. In this respect we aim to present the contributions of non-Euclidean geometry that deeply influenced the young Rueff.

The non-Euclidean metaphor

The contributions of non-Euclidean geometry

During his youth, Rueff read Science and Hypothesis (1902) and The Value of Science (1905) by Henri Poincaré (1854Poincaré ( -1912)). He also became familiar with the geometry developed by Riemann and Lobachevsky. Established for over 2,000 years, Euclidean geometry is still taught in schools today, and is based upon five major postulates designed to demonstrate various proposals in a three-dimensional space (width, height, depth). Here is the fifth postulate: According to this postulate, for any point on a line L, there can pass only one parallel line to the given line A. Euclid originally proved it in these terms:

And that if a straight-line falling across two (other), straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straightlines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side). ( 2007 This intellectual development in geometry coincided with related scientific developments in other areas of knowledge. This is particularly the case in physics, which in the early 1920s saw the work of Albert Einstein, Marcel Grossman (1878[START_REF]The General Theory of Employment, Interest and Money[END_REF]) and David Hilbert (1862-1943), based upon that of James Clerk Maxwell (1831-1879). These developments particularly concerned the move from the Galilean relativity principle, to the theories of special and general relativity. The theory of general relativity arose as a relativistic theory of gravitation, supplanting the theory of gravitation developed by Isaac Newton (1642-1727). Much the same applies regarding research by physicists such as Niels Bohr (1885-1962), Max Planck (1858-1947) and Erwin Schrödinger (1887[START_REF]Chômage et salaires réel en Angleterre dans la période 1919-1925, Séance de la Société d'Économie Sociale du 14 décembre 1925[END_REF] during the first third of the twentieth century, all of whom questioned postulates from classical mechanics in the process of developing what would become quantum mechanics. These intellectual movements in physics are not, in itself, due to geometry, but we should note that the theory of general relativity by Einstein would have been impossible without the use of the differential geometry of Marcel Grossman (including work by Riemann). It would be better, then, to speak about the influence of geometry in physics -and in the same way, we may see it as having an influence in political economy (see the intense debate between John S. Mill (1806-1873) and William S. Jevons (1835-1882)). Hence, we may conclude, such inter-relationships in sciences were numerous at this time, and exceedingly fruitful. Rueff became one of the first French economists to adopt a relativist position and rank each theory according to this geometrical dichotomy4 :

As this ultimate reality is not given, we create it by enunciating a certain number of axioms and definitions which we make the premises of reasoning for deducing the empirical laws. Extending the theory thus constructed, we derive there from new laws which observation may either confirm or invalidate. In the first case, the logical theory remains true. We say that the created causes are also true and that the theory derived from them is a Euclidean theory. When the coincidence of the consequences with observed laws does not take place, the theory remains logical but it is no longer true. We say that it is a non-Euclidean theory. (1929 [1922]: 106) Indeed Rueff only focuses on statistical relations between tools he designed. Rueff built his tools on elements that can be discussed but he was only interested in the interactions between these variables. That is why Rueff based his analysis on wholesale prices. Only accounts the movement of these prices compared with other variables such as the unemployment rate. Only accounts the link between all these variables. Here stands a conception of science and tools which is deeply influenced by advances in non-Euclidean geometry. Indeed it provides the idea that reality, the concrete observation of facts, is not essential anymore to produce knowledge. Hence this kind of geometry is not a priori visible in our plan. However, knowledge are produced and recognized by the scientific community. Then, the imperative of positive observation is no longer relevant in Rueff's views as it paves the way to a conception of reality as inaccessible. Hence "the confirmation of the fact, that real space is nothing more than a coherent system of definitions making the interpretations of observed appearances possible, is found in the existence of non-Euclidean geometries" (1929 [1922]: 32). This is upon this basis that Rueff builds its statistical tools. Then we shall analyse its conception of reality in order to better understand the scope of this approach. [START_REF] Rueff | From the Physical to the Social Sciences[END_REF][START_REF] Rueff | From the Physical to the Social Sciences[END_REF]) allows that sciences can be divided into an empirical and a rational branch: the former collects facts, while the latter reasons on the basis of these facts in order to provide knowledge. Based on this division of labour, the sciences work together by observing the succession of facts and designing laws based upon these successions. The concept of science as it appears in Rueff's work is based on the intellectual approach known as constructivism. Inspired by Immanuel Kant (1724-1804), constructivism is the view that our picture of reality is not reality itself. Rueff holds that the human mind (the rational ego) builds up the sciences with the help of a specific set of tools (the reasoning machine) consisting of syntax, formal logic and mathematical analysis.

The reasoning machine and rational ego

Rueff bases his constructivist view of science upon two principles of human reason: identity and causality. The human mind cannot reason outside of these two principles, which are given a priori and ensure the good conduct of any reasoning. As an affirmation of the relationship between two things using the copula "is" (as in "A is B"), the principle of identity lies at the basis of deductive reasoning and mathematical induction. On the other hand, causality is based upon induction, and links things together in the way which governs the succession -or the concatenation, as Rueff put it -of external realities. Hence, in all being constructed in the same way by this reasoning machine, the sciences only differ according to their object:

We in nowise distinguish the mathematical sciences from the physical or natural, nor the social sciences. We claim that all sciences are of the same type, each comprising an experimental or observational branch which gleans the facts and extracts therefrom empirical laws, and a rational branch which "creates the cause." (1929 [1922]: 22)

The principle of identity allows for observation and deductive reasoning. It is the principle of the justification of mathematical science, paving the way to aggregative rules such as "what is true of the group is true of each individual component thereof" (Ibid.,6). When a thing is dependent on a theory, then, by syllogism, the facts of the same type depend on this theory: the human mind observes that an individual A is a man; he refers to the theory "all men are mortal" and concludes that the individual A is mortal. There is no real knowledge produced here, because our mind is only comparing a fact with a given theory. Here, then, stands the necessity of the principle of non-contradiction. Finally, these two principles underlie the rational and empirical branches of science. The first, necessarily subsequent to the second, is related to the creation of the causes, while the second concerns any given collection of facts that it aims to express under the form of laws.

Before submitting our thought to the principle of causality, it is necessary to formulate this thought by using syntax. Indeed, without using syntax our thought remains free-form and personal. In order to communicate ideas to others, a medium of exchange is necessary; it is this which gives rise to words, phrases and syntax in order to make the interchange of ideas possible. A thought, on this account, is expressed by a proposal that comprises a judgment submitted to the reasoning machine. Each proposal has to be very simple, and its content should appear immediately so as to ensure that it is free from internal contradictions. This principle of non-contradiction only applies to thought and not to syntax, while the latter only concerns the shape of the thought and not its substance. And a thought that does not obey this principle is not a human thought at all, as "it is a fact that only those judgements are 'human' which contain no contradiction [...] for beings such as ourselves, a contradictory judgment is inconceivable" (Ibid.,(10)(11).

In so far as there are false propositions, for instance, that "the Sun revolves around the Earth", this is due to the large number of characteristics in words that are not explicit. Based on the proposal that "a definition is a creative proposition which attaches certain characteristics to a word and by this fact gives it its existence" (Ibid., 13), Rueff argues that the definitions of "Sun", "Moon" and "Earth" are lacking characteristics such as heliocentrism, gravity, centrifugal force and other facts relevant to their interaction. This is why we should take care to keep propositions clear, especially in case of complex judgments. In fact, if a judgment is not compatible with the principle of causality, this is only due to a lack of clarity in our vocabulary.

Here stands the need for a method which enables us to link propositions so that they remain free from contradictions, in such a way as to deliver more complex reasoning. For instance, the reasoning in terms of mass exerting a force that makes a trajectory curved (in the case of gravitation) is certainly a complex judgment, but this is based upon demonstrated and observed principles. These constitute the axioms, "a priori judgments which we are free either to accept or reject" (Ibid.,12). In this respect, axioms enable us to discover new scientific laws: the reasoning machine as a relentless spiral of formal logic starts from two coherent propositions in such a way as to reach a third. Rueff states that this spiral creates nothing in itself, but only reveals reality, which nevertheless remains fundamentally unknown -inaccessible -to the human mind. Indeed, "nothing but a new definition or axiom, incorporated into a new system -i.e., united with a series of propositions already admitted or established, can create new mathematical 'facts' which then need only to be rendered 'explicit'" (Ibid.,14). Once the reasoning is seen as a relentless spiral, it is upstream that we need to bring in new raw material. Rueff accordingly develops a method "which doesn't focus on the object itself, nor on the characteristics of the studied discipline, but rather on its linguistic formulation (Claassen 1967: 38).

The human mind can only work with tools, which are conventions such as methods of measurement or the axioms we have affirmed. In this, by referring to the principle of identity, the aim is to "render intelligible the system of our ever-changing sensations" (1929 [1922]: 7), but it follows that the human mind is what really creates connections of causation. For instance, gravity is only the cause that creates the following effects: the "Moon revolves around the Earth," and "the Earth revolves around the Sun". Therefore "the Law of Causality, which seems to be the expression of physical reality independent of ourselves, owes its existence merely to the effort of our mind to impose that law upon the world" (Ibid., 10). Hence the human mind is not passive, as it apprehends, through this principle of causality, a constant succession of appearances. Prior to the rational branch, the empirical one observes sequences of appearances that are generalized into laws of succession: but this doesn't concern the law of causality, as causes are entirely created by the needs of the human spirit to conform human thought to an appearance of reality:

The role of the human mind is not purely passive in the observation of the laws constituting the empirical branch of science. Man does more than merely record appearances. He determines them to a great extent [...] To say that the empirical laws of our world have existence in themselves is an affirmation without any meaning. (Ibid.,39) Thus, according to the early Rueff, all science creates causes in order to find, via deduction, laws that connect sensory experiences. These views on science are explained by the discovery of non-Euclidean geometry. This profoundly influenced his reasoning on science in general and on political economy in particular. Thus, Rueff defines his variables (unemployment, price, wage) and tries to combine them in order to set out his reasoning. This paves the way to political economy becoming seen as a statistical science (which is the title of his article of 1925a, see also 1961 [1929]). Here we clearly see the links with geometry, as Rueff only deals with the relation between variables and looks for a succession of facts. More precisely, in his work it is possible to identify some elements which prefigure econometrics. Indeed, his work is based upon a permanent relationship between the level of unemployment and wage/price ratio, i.e. a correlation, one of the main purposes of the econometric field. A correlation -a term drawn from biology -would be defined as the intensity in the statistical sense of a link between variables, or numerical statistics, and this is exactly what provides the basis of Rueff's analysis. Obviously, Rueff is still far from the linear regression models and other affine relations that would emerge in future decades, but we must admit that here lie the roots of econometrics. Moreover, having established this strong correlation, Rueff looks for a causal link. Such a link may exist between these variables based upon work by Stamp, consisting in a computing the correlation coefficient between the two variables studied:

This was a quite exceptional correlation in its permanency, and such a correlation permits an affirmation that the two elements under consideration -unemployment and the wage/price ratio -were either dependent on one another due to a causal relationship, or were dependent on a third factor which would be the common cause of variations. (1931a: 213)

Concluding remarks

The study of the relationship between unemployment and the wage/price ratio shows a clear succession of facts. Unemployment in England between the two World Wars is due to the viscosity of this ratio. This is why Rueff is often seen as a liberal who advocates pay cuts and disregards the working classes: but this is a mistake. No matter the level of wage, the sole factor which needs to be accounted for is the way wages are determined, which has to be free and in perfect accord with the rules of the liberal market. This is based upon a structure of reasoning that recalls the axiomatic method of geometry and which demonstrates that there is no need to observe the sensible world to practice science.

The result of these lessons learned from non-Euclidean geometry is that we can study variables and economic quantities (unemployment, wages, prices). The sensible world no longer has a monopoly on the creation of knowledge. It is possible to study variables and their succession, and provide scientific laws upon this basis.

In the end, the conception of reality as it appears in Rueff's work is rather sceptical of the capacity of the human mind. Our minds, being unable to access the real, only create appearances of reality and not reality itself. Where does the production of knowledge come from? How can the human mind formulate any idea? Yet even if reality is inaccessible in itself, as Rueff thinks, it is still possible to build knowledge in such a way as to approach it as closely as possible. The human mind has access to the products of the interaction between thought and its environment, not to the environment itself. In this, language, as well as the reasoning machine and rational ego, prove to be indispensable tools.

Figure n°1 -

 n°1 Figure n°1 -Euclidean postulate n. 5 (the parallel postulate)

  century, brilliant thinkers built a new geometry upon the rejection of this postulate. Although Lobachevsky, Riemann and Bolyai are most often mentioned in connection with the development of a geometry which dispenses with the fifth postulate, it was Gauss who first succeeded in opening the way to non-Euclidean geometry. His results were not published immediately, as they formed part of his correspondence with Heinrich Christian Schumacher (1757-1830) in the 1820s. Being exceptionally difficult to understand, the works of Lobachevsky and Bolyai in the years 1830-1840 remained widely unnoticed. Proper acknowledgment had to wait for more than 20 years, with the publication of the correspondence between Gauss and Schumacher and the translation of Bolyai and Lobachevsky's books into several languages.Non-Euclidean geometry went through a significant process of development, with results provided first in a four-dimensional framework and then for n-dimensions, driven by famous authors such as Poincaré.

Figure 2 -Figure 3 -

 23 Figure 2 -Sphere of parallel lines passing through the same points to the North and South

•

  Beveridge, W.-H., 1909, Unemployment, a Problem of Industry, London: Longman, Green and Co. • Chivvis, C., 2010, The Monetary Conservative: Jacques Rueff and Twentieth-century free market thought, De Kalb: Northern Illinois University Press. • Claassen, E.-M., 1967, "Jacques Rueff comme philosophe des sciences", pp. 37-62, in Claassen, E.-M., 1967, Les Fondements Philosophiques des Systèmes Economiques, Textes de Jacques Rueff et essais rédigés en son honneur, Paris: Payot. • Euclid, 2007, Euclid's Elements of Geometry, The Greek text of J.L. Heiberg (1883-1885) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883-1885, Edited and provided with a modern English translation, by Richard Fitzpatrick, http://www.math.unipd.it/ • Fisher, I., 1926, "Les relations statistiques entre le chômage et le mouvements des prix", Revue Internationale du Travail, 12(6): 846-853. • Frobert L., 2010, "Conventionalism and liberalism in Jacques Rueff's early works (1922 to 1929)", European Journal of History of Economic Thought, 17(3): 439-470. • Hick, J., 1932, The Theory of Wages, London: Mac Millan. • Keynes, J.-M., 1929a, "The German transfer problem", The Economic Journal, 39: 1-17.Table n. 1: Variations of prices, wages, wage/price ratio, and unemployment rate. Figures from Rueff (1925b).

		1	231		309		0.747		1.9
		2	250		324		0.771		1.1
		3	267		314		0.850		1.7
		4	273		284		0.961		5.0
	1921	1 2	276 268		227 201	-44,6	1.251 1.333	+72,1	8.5 20.9	+19,2
		3	244		190		1.284		16
		4	228	-29,5	174		1.310		16
		1	215		162		1.327		16.5
	1922	2 3	202 189		160 157		1.262 1.203		16.4 14.5
		4 1	178 177		156 158		1.141 1.120	-19,6	14.1 13	-8,8
	1923	2 3	177 174		160 156		1.106 1.115		11.2 11.3
		4 1	173 174		161 166	-5,6	1.014 1.048		10.4 8.3
	1924	2 3	177 179	+1,1	164 165		1.079 1.084		7.2 8
	1925	4 1 2 3	179 181 181 180		170 169 160 157		1.052 1.071 1.138 1.153	+12,2	8.8 9.1 10.6 11.3	+3,1
		4	180		153		1.176		11.1

In the French edition of 1922, Rueff uses the term of "principe de causalité" which Herman Green translates as "Law of Causality"[START_REF] Rueff | From the Physical to the Social Sciences[END_REF][START_REF] Rueff | From the Physical to the Social Sciences[END_REF]). We choose to keep the term "principle of causality" except in quotations from this English edition.

Interestingly, Keynes uses the same dichotomy to show that his General Theory is a non-Euclidean one(1936: 16).

Tables

Wages