Resource Allocation in One-dimensional Distributed Service Networks with Applications
Nitish K. Panigrahy, Prithwish Basu, Philippe Nain, Don Towsley, Ananthram Swami, Kevin S. Chan, Kin K. Leung
RResource Allocation in One-dimensional Distributed Service Networks with Applications
Nitish K. Panigrahy a , ∗ , Prithwish Basu b , Philippe Nain c , Don Towsley a , Ananthram Swami d ,Kevin S. Chan d and Kin K. Leung e a University of Massachusetts Amherst, MA 01003, USA b Raytheon BBN Technologies, Cambridge, MA 02138, USA c Inria, 06902 Sophia Antipolis Cedex, France d Army Research Laboratory, Adelphi, MD 20783, USA. e Imperial College London, London SW72AZ, UK.
A R T I C L E I N F O
Keywords :Resource Allocation1-D service networkQueueing TheoryDistributed NetworkDynamic Programming
A B S T R A C T
We consider assignment policies that allocate resources to users, where both resources and usersare located on a one-dimensional line [0 , ∞) . First, we consider unidirectional assignment poli-cies that allocate resources only to users located to their left. We propose the Move to Right( MTR ) policy, which scans from left to right assigning nearest rightmost available resource toa user, and contrast it to the Unidirectional Gale-Shapley (
UGS ) matching policy. While bothpolicies among all unidirectional policies, minimize the expected distance traveled by a request( request distance ), MTR is fairer. Moreover, we show that when user and resource locationsare modeled by statistical point processes, and resources are allowed to satisfy more than oneuser, the spatial system under unidirectional policies can be mapped into bulk service queueingsystems, thus allowing the application of many queueing theory results that yield closed formexpressions. As we consider a case where different resources can satisfy different numbers ofusers, we also generate new results for bulk service queues. We also consider bidirectional poli-cies where there are no directional restrictions on resource allocation and develop an algorithmfor computing the optimal assignment which is more efficient than known algorithms in the lit-erature when there are more resources than users. Numerical evaluation of performance of uni-directional and bidirectional allocation schemes yields design guidelines beneficial for resourceplacement. Finally, we present a heuristic algorithm, which leverages the optimal dynamic pro-gramming scheme for one-dimensional inputs to obtain approximate solutions to the optimalassignment problem for the two-dimensional scenario and empirically yields request distanceswithin a constant factor of the optimal solution.
1. Introduction
The past few years have witnessed significant growth in the use of distributed network analytics involving agilecode, data and computational resources. In many such networked systems, for example, Internet of Things [5], alarge number of computational and storage resources are widely distributed in the physical world. These resourcesare accessed by various end users/applications that are also distributed over the physical space. Assigning users orapplications to resources efficiently is key to the sustained high-performance operation of the system.In some systems, requests are transferred over a network to a server that provides a needed resource. In othersystems, servers are mobile and physically move to the user making a request. Examples of the former type of serviceinclude accessing storage resources over a wireless network to store files and requesting computational resources torun image processing tasks; whereas an example of the latter type of service is the arrival of ride-sharing vehicles tothe user’s location over a road transportation network.Not surprisingly, the spatial distribution of resources and users in the network is an important factor in determiningthe overall performance of the service. A key measure of performance is average request distance , that is average ∗ Corresponding author ∗∗ The material in this paper was presented in part at the IEEE International Symposium on Modeling, Analysis, and Simulation of Computerand Telecommunication Systems (MASCOTS), Rennes, France in 2019 and in Workshop on MAthematical performance Modeling and Analysis(MAMA 2018). [email protected] (N.K. Panigrahy); [email protected] (P. Basu); [email protected] (P. Nain); [email protected] (D. Towsley); [email protected] (A. Swami); [email protected] (K.S. Chan); [email protected] (K.K. Leung)
ORCID (s): We use the terms “users” and “requesters” interchangeably and same holds true for the terms “resources” and “servers”.
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Page 1 of 28 a r X i v : . [ c s . PF ] N ov ne-dimensional Distributed Service Networks distance between a user and its allocated resource/server (where distance is measured on the network). This directlytranslates to latency incurred by a user when accessing the service, which is arguably among the most important criteriain distributed service applications. For example, in wireless networks, signal attenuation is strongly coupled to requestdistance, therefore developing allocation policies to minimize request distance can help reduce energy consumption, animportant concern in battery-operated wireless networks. Another important practical constraint in distributed servicenetworks is service capacity . For example, in network analytics applications, a networked storage device can onlysupport a finite number of concurrent users; similarly, a computational resource can only support a finite number ofconcurrent processing tasks. Likewise, in physical service applications like ride-sharing, a vehicle can pick up a finitenumber of passengers at once.Therefore, a primary problem in such distributed service networks is to efficiently assign each user to a suitableresource so as to minimize average request distance and ensure no resource serves more users than its capacity. If theentire system is being managed by a single administrative entity such as a ride sharing service, or a datacenter networkwhere analytics tasks are being assigned to available CPUs, there are economic benefits in minimizing the averagerequest distance across all (user, resource) pairs, which is tantamount to minimizing the average delay in the system.The general version of this capacitated assignment problem can be solved by modeling it as a minimum cost flow problem on graphs [4] and running the network simplex algorithm [17]. However, if the network has a low-dimensionalstructure and some assumptions about the spatial distributions of users and resources hold, more efficient methods canbe developed.In this paper, we consider two one-dimensional network scenarios that motivate the study of this special case ofthe user-to-resource assignment problem.The first scenario is ride-hailing on a one-way street where vehicles move right to left. If the vehicles of a ride-sharing company are distributed along the street at a certain time, and users equipped with smartphone ride-hailingapps request service, the system attempts to assign vehicles with spare capacity located towards the right of the usersso as to minimize average “pick up" distance. Abadi et al. [1] introduced this problem and presented a policy known asUnidirectional Gale-Shapley matching ( UGS ) minimize average pick up distance. In this policy, all users concurrentlyemit rays of light toward their right and each user is matched with the vehicle that first receives the emitted ray. Whilethe well-known Gale-Shapley matching algorithm [9] matches user-resource pairs that are mutually nearest to eachother, its unidirectional variant, UGS, matches a user to the nearest resource on its right. Note that, this one-dimensionalnetwork setting also applies to vehicular wireless ad-hoc networks on a one-lane roadway [11, 14] , where users are invehicles and servers are attached to fixed infrastructure such as lamp posts. Users attempt to allocate their computationtasks over the wireless network to servers located to their right so that they can retrieve the results with little effortwhile driving by.In this paper, we propose another policy “Move to Right” policy (or MTR ) which has the same “expected distancetraveled by a request” ( request distance ) as UGS but has a lower variance.
MTR sequentially allocates users to thegeographically nearest available vehicle located to his/her right. When user and resource locations are modeled bystatistical point processes the one-dimensional unidirectional space behaves similar to time and notions from queueingtheory can be applied. In particular, when user and vehicle locations are modeled by independent Poisson processes,average request distance can be characterized in closed form by considering inter-user and inter-server distances asparameters of a bulk service
M/M/1 queue where the bulk service capacity denotes the maximum number of usersthat can be handled by a server. We equate request distance in the spatial system to the expected sojourn time in thecorresponding queuing model . This natural mapping allows us to use well-known results from queueing theory andin some cases to propose new queueing theoretic models to characterize request distances for a number of interestingsituations beyond M/M/1 queues.A natural extension to our spatial framework is to consider more general communication costs associated with eachresource allocation. Assuming communication cost for each allocation is a function of request distance, we provideclosed form expressions for the expected communication cost for specific user-server distributions and specific servercapacities.The second scenario involves a convoy of vehicles traveling on a one-dimensional space, for example, trucks on ahighway or boats on a river. Some vehicles have expensive camera sensors (image/video) but have inadequate com- We rename queue matching defined in [1] as Unidirectional Gale-Shapley Matching to avoid overloading the term queue . Furthermore, [11] confirms that vehicle location distribution on the streets in Central London can be closely approximated by a Poissondistribution. Sojourn time is the sum of waiting and service times in a queue.
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Page 2 of 28ne-dimensional Distributed Service Networks putational storage or processing power. On the other hand, cheap storage and processing is easily available on severalother vehicles. The cameras periodically take photos/videos as they move through space and want them processed/ stored. In such case, bidirectional assignment schemes are more suitable. Since no directionality restrictions areimposed on the allocation algorithms, computing the optimal assignment is not as simple as in the unidirectional case.We explore the special structure of the one-dimensional topology to develop an optimal algorithm that assigns aset of requesters 𝑅 to a set of resources 𝑆 such that the total assignment cost is minimized. This problem has beenrecently solved for | 𝑅 | = | 𝑆 | [7]. However, we are interested in the case when | 𝑅 | < | 𝑆 | . We propose a dynamicProgramming based algorithm which solves this case with time complexity 𝑂 ( | 𝑅 | ( | 𝑆 | − | 𝑅 | + 1)) . Note that otherassignment algorithms in literature such as the Hungarian primal-dual algorithm and Agarwal’s variant [3] have timecomplexities 𝑂 ( | 𝑅 | ) and 𝑂 ( | 𝑅 | 𝜖 ) respectively and assume | 𝑅 | = | 𝑆 | for general and Euclidean distance measures.We leverage the optimal dynamic programming scheme for one-dimensional inputs to obtain approximate solutionsto the optimal assignment problem for the two-dimensional scenario where users and servers are located on the two-dimensional plane, ℝ . More precisely, we embed the points denoting 𝑅, 𝑆 ⊂ ℝ into new locations in ℝ such that thedistances between a user and its nearest servers are approximately preserved. Our approximation algorithm empiricallyyields request distances within a constant factor of the optimal solution with 𝑂 ( | 𝑅 | ) time complexity.Our contributions are summarized below:1. Analysis of simple unidirectional allocation policies MTR and
UGS yielding closed form expressions for meanrequest distance.• When inter-requester and inter-resource distances are exponentially distributed, we model unidirectionalpolicies as a bulk service M/M/1 queue.• When inter-requester distances are generally distributed but the inter-resource distances are exponentiallydistributed, we model the situation using an accessible batch service G/M/1 queue.• When inter-requester distances are exponentially distributed but inter-resource distances are generally dis-tributed, we model the spatial system as an accessible batch service M/G/1 queue with the first batch havingexceptional service time. To the best of our knowledge this system has not been studied previously in thequeueing theory literature.• We include several generalizations of our framework. In the first place we discuss a simulation drivenconjecture for evaluating request distance for general distance distributions under heavy traffic. We alsoinvestigate the heterogeneous server capacity scenario where server capacity is a random variable and tothe best of our knowledge this system has not been studied previously in the queueing theory literature. Wederive expressions for expected request distance when servers have infinite capacity. We include commu-nication cost associated with each resource allocation and provide a closed form expression for expectedcommunication cost for some specific scenarios. Finally we extend our framework to compute expectedrequest distance for the case where each user requests two resources residing in two different set of servers,by mapping it to a two queue fork-join system.2. A novel algorithm for optimal (bidirectional) assignment with time complexity 𝑂 ( | 𝑅 | ( | 𝑆 | − | 𝑅 | + 1)) .3. A numerical and simulation study of different assignment policies: UGS , MTR, bi-directional heuristic alloca-tion policies (Gale-Shapley and Nearest Neighbor) and the optimal policy.4. A heuristic based approximate solution to the optimal assignment problem for the two-dimensional scenariowith an empirically observed constant factor approximation of the optimal solution.The paper is organized as follows. The next section discusses related work. Section 3 contains technical prelim-inaries. We show the equivalence of UGS and MTR w.r.t expected request distance in Section 4, and present resultsassociated with the case when servers are Poisson distributed in Section 5. In Section 6, we develop formulationsfor expected request distance when either user or server placements are described by Poisson processes. We includesome generalizations of our framework such as analysis under general distance distributions, results for heterogeneousserver capacity and uncapacitated allocation in Section 7. The optimal bidirectional allocation strategy is presented inSection 8. We compare the performance of various local allocation strategies in Section 9. In Section 10, we extendour one-dimensional framework to solve two-dimensional problem. We conclude the paper in Section 11. Panigrahy et al.:
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2. Related Work
Poisson Matching:
Holroyd et al. [12] first studied translation invariant matchings between two 𝑑 -dimensional Poissonprocesses with equal densities. Their primary focus was obtaining upper and lower bounds on expected matchingdistance for stable matchings. Abadi et al. [1] introduced “Unidirectional Gale-Shapley” matching ( UGS ) and derivedbounds on the expected matching distance for stable matchings between two one-dimensional Poisson processes withdifferent densities. In this paper, we propose another unidirectional allocation policy: “Move To Right” policy (
MTR )and provide explicit expressions for the expected matching distance for both MTR and UGS when either requesters orservers are distributed according to a renewal process and the according to a Poisson process.
Exceptional Queueing Systems and Accessible Batches:
Welch et al. [22] first studied an M/G/1 queue where a cus-tomer arriving when the server is idle has a different service time than the others. Bulk service M/G/1 queues has beenstudied in [6]. Authors in [10] analyzed a bulk service G/M/1 queue with accessible or non-accessible batches wherean accessible batch is considered to be a batch in service allowing subsequent arrivals, while the service is on. Inthis work, we model the spatial system using an accessible batch service queue with the first batch having exceptionalservice time. To the best of our knowledge this system has not been studied previously in queueing theory literature.
Euclidean Bipartite Matching:
The optimal user-server assignment problem can be modeled as a minimum-weightmatching on a weighted bipartite graph where weights on edges are given by the Euclidean distances between thecorresponding vertices [15]. Well-known polynomial time solutions exist for this problem, such as the modified Hun-garian algorithm proposed by Agarwal et al. [3] with a running time of 𝑂 ( | 𝑅 | 𝜖 ) , where | 𝑅 | is the total number ofusers. In the case of an equal number of users and servers, the optimal user-server assignment on a real line is known[7]. In this paper, we consider the case when there are fewer users than servers.
3. Technical Preliminaries
Consider a set of users 𝑅 and a set of servers 𝑆 . Each user makes a request that can be satisfied by any server.Assume that each server 𝑗 ∈ 𝑆 has capacity 𝑐 𝑗 ∈ ℤ + corresponding to the maximum number of requests that it canprocess. Suppose users and servers are located on a line . Formally, let 𝑟 ∶ 𝑅 → and 𝑠 ∶ 𝑆 → be the locationfunctions for users and servers, respectively, such that a distance 𝑑 ( 𝑟, 𝑠 ) is well defined for all pairs ( 𝑟, 𝑠 ) ∈ 𝑅 × 𝑆 .Initially we assume that all servers have equal capacities i.e. 𝑐 𝑗 = 𝑐 ∀ 𝑗 ∈ 𝑆. Later in Section 7.2 we extend our analysisto a case in which server capacities are integer random variables.
Let ≤ 𝑟 ≤ 𝑟 ≤ ⋯ represent user locations and ≤ 𝑠 ≤ 𝑠 ≤ ⋯ be the server locations. Let 𝑋 𝑗 = 𝑠 𝑗 − 𝑠 𝑗 −1 , 𝑗 ≥ , 𝑠 = 0 , denote the inter-server distances and 𝑌 𝑖 = 𝑟 𝑖 − 𝑟 𝑖 −1 , 𝑖 ≥ , 𝑟 = 0 , the inter-user distances. Weassume { 𝑋 𝑗 } 𝑗 ≥ to be a renewal process with cumulative distribution function (cdf) ℙ ( 𝑋 𝑗 ≤ 𝑥 ) = 𝐹 𝑋 ( 𝑥 ) . (1)We also assume { 𝑌 𝑖 } 𝑖 ≥ to be a renewal process with cdf 𝐹 𝑌 ( 𝑥 ) , i.e., ℙ ( 𝑌 𝑖 ≤ 𝑥 ) = 𝐹 𝑌 ( 𝑥 ) . (2)We denote 𝛼 𝑋 = 1∕ 𝜇 and 𝜎 𝑋 to be the mean and variance associated with 𝐹 𝑋 . Similarly let 𝛼 𝑌 = 1∕ 𝜆 and 𝜎 𝑌 be themean and variance associated with 𝐹 𝑌 . We let 𝜌 = 𝜆 ∕ 𝜇 and assume that 𝜌 < 𝑐 . Denote by 𝐹 ∗ 𝑋 ( 𝑠 ) = ∫ ∞0 𝑒 − 𝑠𝑥 𝑑𝐹 𝑋 ( 𝑥 ) and 𝐹 ∗ 𝑌 ( 𝑠 ) the Laplace-Stieltjes transform ( LST ) of 𝐹 𝑋 and 𝐹 𝑌 with 𝑠 ≥ . In our paper, we consider various inter-server and inter-user distance distributions, including exponential, deter-ministic, uniform and hyperexponential.
One of our goals is to analyze the performance of various request allocation policies using expected request distanceas a performance metric. We define various allocation policies as follows.•
Unidirectional Gale-Shapley (
UGS ): In UGS, each user simultaneously emits a ray to their right. Once the rayhits an unallocated server 𝑠 , the user is allocated to 𝑠 . Panigrahy et al.:
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Figure 1:
Allocation of users to servers on the one-dimensional network. Top: UGS, Bottom: MTR allocation policy. • Move To Right (
MTR ): In MTR, starting from the left, each user is allocated sequentially to the nearest availableserver to its right.•
Nearest Neighbor ( NN ) [21]: In this matching, starting from the left, each user is allocated sequentially to thenearest available server. This policy can be viewed as the bidirectional version of MTR policy.•
Gale-Shapley ( GS ) [9]: In this matching, each user selects the nearest server and each server selects its nearestuser. Remove reciprocating pairs, and continue.•
Optimal Matching:
This matching minimizes average request distance among all feasible allocation policies.
4. Unidirectional Allocation Policies
In this Section, we establish the equivalence of UGS and MTR w.r.t number of requests that traverse a point andexpected request distance. Define 𝑁 𝑃𝑥 and 𝐷 𝑃𝑖 to be random variables for the number of requests that traverse point 𝑥 ∈ and distance between user 𝑖 and its allocated server under policy 𝑃 , respectively. Thus 𝑁 𝑈𝑥 and 𝑁 𝑀𝑥 denote thenumber of requests that traverse point 𝑥 ∈ under UGS and MTR, respectively, as shown in Figure 1. Consider thefollowing definition of busy cycle in a service network. Definition 1.
A busy cycle for a policy P is an interval 𝐼 = [ 𝑎, 𝑏 ] ⊂ such that ∃ 𝑖, 𝑗 with 𝑟 𝑖 = 𝑎, 𝑠 𝑗 = 𝑏 for which 𝑁 𝑃𝑥 > , ∀ 𝑥 ∈ 𝐼 and 𝑁 𝑃𝑥 = 0 for 𝑥 = 𝑎 − 𝜖 and 𝑥 = 𝑏 + 𝜖 with 𝜖 being an infinitesimal positive value. We have the following theorem.
Theorem 1. 𝑁 𝑈𝑥 = 𝑁 𝑀𝑥 , 𝑥 ≥ . Proof.
Due to the unidirectional nature of matching, both UGS and MTR have the same set of busy cycles. Denote as the set of all busy cycles in the service network. In the case when 𝑥 ∈ ⧵ ⋃ 𝐼 ∈ 𝐼 we already have 𝑁 𝑈𝑥 = 𝑁 𝑀𝑥 = 0 . Let us now consider a busy cycle 𝐼 𝑈 = [ 𝑎 𝑈 , 𝑏 𝑈 ] under UGS policy. Let 𝑥 ∈ 𝐼 𝑈 . Let 𝐿 𝑈𝑥,𝑅 = | { 𝑟 𝑖 | 𝑎 𝑈 ≤ 𝑟 𝑖 ≤ 𝑥 } | and 𝐿 𝑈𝑥,𝑆 = | { 𝑠 𝑗 | 𝑎 𝑈 ≤ 𝑠 𝑗 ≤ 𝑥 } | . 𝑁 𝑈𝑥 = 𝐿 𝑈𝑥,𝑅 − 𝐿 𝑈𝑥,𝑆 . Similarly define 𝐿 𝑀𝑥,𝑅 and 𝐿 𝑀𝑥,𝑆 for MTR policy. Clearly 𝑁 𝑀𝑥 = 𝐿 𝑀𝑥,𝑅 − 𝐿 𝑀𝑥,𝑆 . As both policies have the same set of busy cycles we have 𝐿 𝑈𝑥,𝑅 = 𝐿 𝑀𝑥,𝑅 and 𝐿 𝑈𝑥,𝑆 = 𝐿 𝑀𝑥,𝑆 . Thuswe get 𝑁 𝑈𝑥 = 𝑁 𝑀𝑥 , 𝑥 ∈ ℝ + , (3) Corollary 1. 𝔼 [ 𝐷 𝑈 ] = 𝔼 [ 𝐷 𝑀 ] i.e. the expected request distances are the same for both UGS and MTR under steadystate. Panigrahy et al.:
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Distribution Parameters 𝐹 𝑋 ( 𝑥 ) 𝐹 𝑋 ( 𝑥 ) 𝐹 𝑋 ( 𝑥 ) ( 𝑥 ) ( 𝑥 ) ( 𝑥 ) Exponential 𝜇 : rate 𝑒 − 𝜇𝑥 𝜆 [ 𝑒 − 𝜆𝑥 ] − 𝜆 + 𝜇 [ 𝑒 −( 𝜆 + 𝜇 ) 𝑥 ] Uniform 𝑏 ∶ maximum value 𝑥 ∕ 𝑏, ≤ 𝑥 ≤ 𝑏 𝜆 𝑏 [ 𝑒 − 𝜆𝑏 ] − 𝑒 − 𝜆𝑥 𝜆 Deterministic 𝑑 ∶ constant , 𝑥 ≥ 𝑑 𝑒 − 𝜆𝑑 − 𝑒 − 𝜆𝑥 𝜆 Hyper 𝑙 : order 𝑙 ∑ 𝑗 =1 𝑝 𝑗 𝑒 − 𝜇 𝑗 𝑥 𝜆 [ 𝑒 − 𝜆𝑥 ] − 𝑙 ∑ 𝑗 =1 𝑝 𝑗 𝜆 + 𝜇 𝑗 [ 𝑒 −( 𝜆 + 𝜇 𝑗 ) 𝑥 ] -exponential 𝑝 𝑗 ∶ phase probability 𝜇 𝑗 ∶ phase rate Table 1
Properties of specific inter-server distance distributions.
Proof.
Under steady state both 𝑁 𝑈𝑥 and 𝑁 𝑀𝑥 converge to a random variable. Applying Little’s law we have 𝔼 [ 𝐷 𝑈 ] = 𝔼 [ 𝐷 𝑀 ] . Remark 1.
Note that Theorem 1 applies to any inter-server or inter-user distance distribution. It also applies to thecase where servers have capacity 𝑐 > . Remark 2.
Although MTR and UGS are equivalent w.r.t. the expected request distance, MTR tends to be fairer, i.e.,has low variance w.r.t. request distance.
5. Unidirectional Poisson Matching
In this section, we characterize request distance statistics under unidirectional policies when both users and serversare distributed according to two independent Poisson processes. We first analyze MTR as follows.
Under this allocation policy, the service network can be modeled as a bulk service M/M/1 queue. A bulk serviceM/M/1 queue provides service to a group of 𝑐 or fewer customers. The server serves a bulk of at most 𝑐 customerswhenever it becomes free. Also customers can join an existing service if there is room which is an example of accessiblebatch. In Section 6 we describe the notion of accessible batches in greater detail. The service time for the groupis exponentially distributed and customer arrivals are described by a Poisson process. The distance between twoconsecutive users in the service network can be thought of as inter-arrival time between customers in the bulk serviceM/M/1 queue. The distance between two consecutive servers maps to a bulk service time.Having established an analogy between the service network and the bulk service M/M/1 queue, we now define thestate space for the service network. Consider the definition of 𝑁 𝑥 as the number of requests that traverse point 𝑋 ∈ 𝐿 under MTR. In steady state, 𝑁 𝑥 converges to a random variable 𝑁 provided 𝜆 < 𝑐𝜇 . Let 𝜋 𝑘 denote Pr [ 𝑁 = 𝑘 ] with 𝑘 ≥ .Following the procedure in Section 4.2.1 of [20], we obtain the steady state probability vector 𝜋 = [ 𝜋 𝑖 , 𝑖 ≥ . Inthe service network, request distance corresponds to the sojourn time in the bulk service M/M/1 queue. By applyingLittle’s formula, we obtain the following expression for the expected request distance 𝔼 [ 𝐷 ] = 𝑟 𝜆 (1 − 𝑟 ) , (4)where 𝑟 is the only root in the interval (0 , of the following equation (with 𝑟 as the variable) 𝜇𝑟 𝑐 +1 − ( 𝜆 + 𝜇 ) 𝑟 + 𝜆 = 0 . (5) It is well known in queueing theory that among all service disciplines the variance of the waiting time is minimized under FCFS policy forPoisson arrivals and exponential service times [13]. In Section 5 we show that MTR maps to a temporal FCFS queue. We drop the superscript ( 𝑀 ) for brevity. Panigrahy et al.:
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Page 6 of 28ne-dimensional Distributed Service Networks s j r i r i+1 r i+2 s j+1 s j+2 Busy Cycle X j+1 X j+2 Z j+1 ServersUsers
Figure 2:
Allocation of users to servers under MTR policy. 𝑐 = 1 When 𝑐 = 1 , 𝑟 = 𝜌 is a solution of (5). Thus we can evaluate the expected request distance as 𝔼 [ 𝐷 ] = 𝜌𝜆 (1 − 𝜌 ) = 1 𝜇 − 𝜆 . (6)Note that, when server capacity is one, the service network can be modeled as an M/M/1 queue. In such a case, (6) isthe mean sojourn time for an M/M/1 queue. When both users and servers are Poisson distributed and servers have unit capacity, the request distance in UGS hasthe same distribution as the busy cycle in the corresponding Last-Come-First-Served Preemptive-Resume (
LCFS-PR )queue having the density function [1] 𝑓 𝐷 𝑈 ( 𝑥 ) = 1 𝑥 √ 𝜌 𝑒 ( 𝜆 + 𝜇 ) 𝑥 𝐼 (2 𝑥 √ 𝜆𝜇 ) , 𝑥 > , (7)where 𝜌 = 𝜆 ∕ 𝜇 and 𝐼 is the modified Bessel function of the first kind. Thus the expected request distance is equivalentto the average busy cycle duration in a LCFS-PR queue given by 𝜇 − 𝜆 ) [1].When servers have capacities 𝑐 > it is difficult to characterize the expected request distance explicitly. However,by Theorem 1, the expected request distance under UGS is the same as that of MTR given by (4).
6. Unidirectional General Matching
We now derive expressions for the expected request distance when either users or servers are distributed accordingto a Poisson process and the other by renewal process.
We discuss the notion of exceptional service and accessible batches applicable to our service network as follows.Consider a service network with 𝑐 = 2 as shown in Figure 2. Consider a user 𝑟 𝑖 . Let 𝑠 𝑗 be the server immediately tothe left of 𝑟 𝑖 . We assume all users prior to 𝑟 𝑖 have already been allocated to servers { 𝑠 𝑘 , ≤ 𝑘 ≤ 𝑗 } . MTR allocatesboth 𝑟 𝑖 and 𝑟 𝑖 +1 to 𝑠 𝑗 +1 and allocates 𝑟 𝑖 +2 to 𝑠 𝑗 +2 . We denote [ 𝑟 𝑖 , 𝑠 𝑗 +2 ] as a busy cycle of the service network. We havethe following queueing theory analogy.User 𝑟 𝑖 can be thought of as the first customer in a queueing system that initiates a busy period while 𝑟 𝑖 +1 seesthe system busy when it arrives. Because only 𝑟 𝑖 is in service at the arrival of 𝑟 𝑖 +1 , 𝑟 𝑖 +1 enters service with 𝑟 𝑖 and thetwo customers form a batch of size 2. and depart at time 𝑠 𝑗 +1 . This is an example of an accessible batch [10]. Anaccessible batch admits subsequent arrivals, while the service is on, until the server capacity 𝑐 is reached.The service time for the batch, 𝑟 𝑖 , 𝑟 𝑖 +1 , is described by the random variable 𝑍 𝑗 +1 which is different or exceptional when compared to service times of successive batches such as the one consisting of 𝑟 𝑖 +2 . The service time for thesecond batch is 𝑋 𝑗 +2 . Note that, 𝑍 𝑗 +1 only depends on 𝑋 𝑗 +2 and 𝑌 𝑖 +2 . Thus when either 𝑋 𝑗 +2 or 𝑌 𝑖 +2 is describedby a Poisson process and the other by renewal process, 𝑍 𝑗 +1 converges to a random variable 𝑍 under steady stateconditions. Denote 𝐹 𝑍 ( 𝑥 ) and 𝑓 𝑍 ( 𝑥 ) as the distribution and density functions for the random variable 𝑍 . Thus the Panigrahy et al.:
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Page 7 of 28ne-dimensional Distributed Service Networks service network can be mapped to an exceptional service with accessible batches queueing (
ESABQ ) model. We for-mally define ESABQ as follows.
ESABQ:
Consider a queueing system where customers are served in batches of maximum size 𝑐 . A customer enteringthe queue and finding fewer than 𝑐 customers in the system joins the current batch and enters service at once, otherwiseit joins a queue. After a batch departs leaving 𝑘 customers in the buffer, min( 𝑐, 𝑘 ) customers form a batch and enterservice immediately. There are two different service times cdfs, 𝐹 𝑍 ( 𝑥 ) (exceptional batch) with mean 𝛼 𝑍 = 1∕ 𝜇 𝑍 and 𝐹 𝑋 ( 𝑥 ) (ordinary batch) with mean 𝛼 𝑋 = 1∕ 𝜇 . A batch is exceptional if its oldest customer entered an empty system,otherwise it is a regular batch. When the service time expires, all customers in the server depart at once, regardlessof the nature of the batch (exceptional or regular). 𝐹 𝑍 ( 𝑥 ) In this Section, we compute explicit expressions for the distribution function 𝐹 𝑍 ( 𝑥 ) applicable to our service net-work. When 𝐹 𝑋 ( 𝑥 ) ∼ 𝐹 𝑋 ( 𝑥 ) ∼ 𝐹 𝑋 ( 𝑥 ) ∼ Expo ( 𝜇 )( 𝜇 )( 𝜇 ) : In this case, we invoke the memoryless property of the exponential distribution 𝐹 𝑋 . Thusthe exceptional distribution, 𝐹 𝑍 , is 𝐹 𝑍 ( 𝑥 ) = 𝐹 𝑋 ( 𝑥 ) = 1 − 𝑒 − 𝜇𝑥 , 𝑥 ≥ . (8) When 𝐹 𝑌 ( 𝑥 ) ∼ 𝐹 𝑌 ( 𝑥 ) ∼ 𝐹 𝑌 ( 𝑥 ) ∼ Expo ( 𝜆 )( 𝜆 )( 𝜆 ) : Using the memoryless property of 𝐹 𝑌 , 𝐹 𝑍 can be computed as 𝐹 𝑍 ( 𝑥 ) = Pr ( 𝑋 − 𝑌 < 𝑥 | 𝑌 < 𝑋 ) = Pr ( 𝑋 − 𝑌 < 𝑥 | 𝑋 − 𝑌 >
0) = Pr ( 𝑋 − 𝑌 < 𝑥 ) − Pr ( 𝑋 − 𝑌 < Pr ( 𝑋 − 𝑌 < 𝐷 𝑋𝑌 ( 𝑥 ) − 𝐷 𝑋𝑌 (0)1 − 𝐷 𝑋𝑌 (0) , 𝑥 ≥ , (9) where 𝐷 𝑋𝑌 ( 𝑥 ) is the distribution of the random variable 𝑋 − 𝑌 (also known as difference distribution). 𝐷 𝑋𝑌 ( 𝑥 ) canbe expressed as 𝐷 𝑋𝑌 ( 𝑥 ) = Pr ( 𝑋 − 𝑌 ≤ 𝑥 ) = ∫ ∞0 Pr ( 𝑋 − 𝑦 ≤ 𝑥 ) Pr ( 𝑌 = 𝑦 ) 𝑑𝑦 = ∫ ∞0 𝐹 𝑋 ( 𝑥 + 𝑦 ) 𝜆𝑒 − 𝜆𝑦 𝑑𝑦 = ∫ ∞ 𝑥 𝐹 𝑋 ( 𝑧 ) 𝜆𝑒 − 𝜆 ( 𝑧 − 𝑥 ) 𝑑𝑧 = 𝜆𝑒 𝜆𝑥 [ ∫ ∞0 𝐹 𝑋 ( 𝑧 ) 𝑒 − 𝜆𝑧 𝑑𝑧 − ∫ 𝑥 𝐹 𝑋 ( 𝑧 ) 𝑒 − 𝜆𝑧 𝑑𝑧 ] = 𝜆𝑒 𝜆𝑥 [ ( 𝐹 𝑋 ) − ( 𝑥 ) ] , (10) where is the Laplace Transform operator on the function 𝐹 𝑋 and ( 𝑥 ) is denoted by ( 𝑥 ) = ∫ 𝑥 𝐹 𝑋 ( 𝑧 ) 𝑒 − 𝜆𝑧 𝑑𝑧 Clearly (0) = 0 . Thus combining (9) and (10) yields 𝐹 𝑍 ( 𝑥 ) = 𝜆𝑒 𝜆𝑥 [ ( 𝐹 𝑋 ) − ( 𝑥 ) ] − 𝜆 ( 𝐹 𝑋 )1 − 𝜆 ( 𝐹 𝑋 ) , (11) 𝑓 𝑍 ( 𝑥 ) = 𝜆 𝑒 𝜆𝑥 [ ( 𝐹 𝑋 ) − ( 𝑥 ) ] − 𝜆𝐹 𝑋 ( 𝑥 )1 − 𝜆 ( 𝐹 𝑋 ) , (12) 𝛼 𝑍 = ∫ ∞0 𝑥𝑓 𝑍 ( 𝑥 ) 𝑑𝑥, 𝜎 𝑍 = [ ∫ ∞0 𝑥 𝑓 𝑍 ( 𝑥 ) 𝑑𝑥 ] − 𝛼 𝑍 . (13)Expressions for ( 𝑥 ) are presented in Table 1. We can evaluate ( 𝐹 𝑋 ) by setting ( 𝐹 𝑋 ) = 𝐵 (∞) . Detailedderivations are relegated to Appendix 13.1.
Panigrahy et al.:
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GRPS ) From our discussion in Section 6.1.1, it is clear that when servers are distributed according to a Poisson process,the exceptional service time distribution equals the regular batch service time distribution. In such a case we have thefollowing queueing model.
Under GRPS, inter-arrival times and batch service times are, respectively, arbitrarily and exponentially distributed.Before initiating a service, a server finds the system in any of the following conditions. (i) ≤ 𝑛 ≤ 𝑐 − 1 and (ii) 𝑛 ≥ 𝑐. Here 𝑛 is the number of customers in the waiting buffer. For case (i) the server provides service to all 𝑛 customersand admits subsequent arrivals until 𝑐 is reached. For case (ii) the server takes 𝑐 customers with no admission forsubsequent customers arriving within its service time. In such a case ESABQ can directly be modeled as a special case of a renewal input bulk service queue withaccessible and non-accessible batches proposed in [10] with parameter values 𝑎 = 1 and 𝑑 = 𝑏 = 𝑐. Let 𝑁 𝑠 and 𝑁 𝑞 denote random variables for numbers of customers in the system and in the waiting buffer respectively for ESABQunder GRPS. We borrow the following definitions from [10]. 𝑃 𝑛, = Pr[ 𝑁 𝑠 = 𝑛 ]; 0 ≤ 𝑛 ≤ 𝑐 − 1 , 𝑃 𝑛, = Pr[ 𝑁 𝑞 = 𝑛 ]; 𝑛 ≥ . (14)Using results from [10] we obtain the following expressions for equilibrium queue length probabilities. 𝑃 , = 𝐶𝜇 [ 𝑟 𝑐 −10 − 𝑟 𝑐 𝑟 𝑐 + 1 𝑟 − 1 ] , 𝑃 𝑛, = 𝐶𝑟 𝑛 −10 (1 − 𝑟 ) 𝜇 (1 − 𝑟 𝑐 ) ; 𝑛 ≥ , (15)where < 𝑟 < is the real root of the equation 𝑟 = 𝐹 ∗ 𝑌 ( 𝜇 − 𝜇𝑟 𝑐 ) and 𝐶 is the normalization constant given by 𝐶 = 𝜆 [ 𝜔 𝑐 𝜔 + 11 − 𝑟 − 𝜔 ( 𝑟 − 𝐹 ∗ 𝑌 ( 𝜇 )) 𝑟 𝑐 (1 − 𝑟 𝜔 ) ( 𝑟 𝑐 𝑟 − 𝑟 𝑐 −10 𝑤 𝑐 𝑤 )] −1 , (16) with 𝜔 = 1∕ 𝐹 ∗ 𝑌 ( 𝜇 ) . We then derive the expected queue length as 𝔼 [ 𝑁 𝑞 ] = ∞ ∑ 𝑛 =0 𝑛𝑃 𝑛, = ∞ ∑ 𝑛 =1 𝑛 𝐶𝑟 𝑛 −10 (1 − 𝑟 ) 𝜇 (1 − 𝑟 𝑐 ) = 𝐶 (1 − 𝑟 ) 𝜇 (1 − 𝑟 𝑐 ) ∞ ∑ 𝑛 =1 𝑛𝑟 𝑛 −10 = 𝐶𝜇 (1 − 𝑟 𝑐 )(1 − 𝑟 ) . (17)Applying Little’s law and considering the analogy between our service network and ESABQ we obtain the followingexpression for the expected request distance. 𝔼 [ 𝐷 ] = 𝐶𝜆𝜇 (1 − 𝑟 𝑐 )(1 − 𝑟 ) + 1 𝜇 . (18) PRGS ) As discussed in Section 6.1.1, if servers are placed on a -d line according to a renewal process with requests beingPoisson distributed, the service time distribution for the first batch in a busy period differs from those of subsequentbatches. Below we derive expressions for queue length distribution and expected request distance for ESABQ underPRGS. We use a supplementary variable technique to derive the queue length distribution for ESABQ under PRGS asfollows.Let 𝐿 ( 𝑡 ) be the number of customers at time 𝑡 ≥ , 𝑅 ( 𝑡 ) the residual service time at time 𝑡 ≥ (with 𝑅 ( 𝑡 ) = 0 if 𝐿 ( 𝑡 ) = 0 ), and 𝐼 ( 𝑡 ) the type of service at time 𝑡 ≥ with 𝐼 ( 𝑡 ) = 1 (resp. 𝐼 ( 𝑡 ) = 2 ) if exceptional (resp. ordinary)service time.Let us write the Chapman-Kolmorogov equations for the Markov chain {( 𝐿 ( 𝑡 ) , 𝑅 ( 𝑡 ) , 𝐼 ( 𝑡 )) , 𝑡 ≥ .For 𝑡 ≥ , 𝑛 ≥ , 𝑥 > , 𝑖 = 1 , define 𝑝 𝑡 ( 𝑛, 𝑥 ; 𝑖 ) = ℙ ( 𝐿 ( 𝑡 ) = 𝑛, 𝑅 ( 𝑡 ) < 𝑥, 𝐼 ( 𝑡 ) = 𝑖 ) and 𝑝 𝑡 (0) = ℙ ( 𝐿 ( 𝑡 ) = 0) . The normalization constant 𝐶 derived in [10] is incorrect. The correct constant for our case is given in (16). Panigrahy et al.:
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Page 9 of 28ne-dimensional Distributed Service Networks
Also, define for 𝑥 > , 𝑖 = 1 , , 𝑝 ( 𝑛, 𝑥 ; 𝑖 ) = lim 𝑡 → ∞ 𝑝 𝑡 ( 𝑛, 𝑥 ; 𝑖 ) and 𝑝 (0) = lim 𝑡 → ∞ 𝑝 𝑡 (0) . By analogy with the analysis for the M/G/1 queue we get 𝜕𝜕𝑡 𝑝 𝑡 (0) = − 𝜆𝑝 𝑡 (0) + 𝑐 ∑ 𝑘 =1 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑘,
0; 1) + 𝑐 ∑ 𝑘 =1 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑘,
0; 2) , so that, by letting 𝑡 → ∞ , 𝜆𝑝 (0) = 𝑐 ∑ 𝑘 =1 ( 𝜕𝜕𝑥 𝑝 ( 𝑘,
0; 1) + 𝜕𝜕𝑥 𝑝 ( 𝑘,
0; 2) ) . (19)With further simplification (See Appendix 13.2.1), for 𝑛 ≥ , 𝑥 > we get 𝜕𝜕𝑥 𝑔 ( 𝑛, 𝑥 ) − 𝜆𝑔 ( 𝑛, 𝑥 ) − 𝜕𝜕𝑥 𝑔 ( 𝑛,
0) + 𝜆𝑔 ( 𝑛 − 1 , 𝑥 ) ( 𝑛 ≥
2) + 𝜆𝑝 (0) 𝐹 𝑍 ( 𝑥 ) ( 𝑛 = 1) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑔 ( 𝑛 + 𝑐,
0) = 0 , (20)where 𝑔 ( 𝑛, 𝑥 ) = 𝑝 ( 𝑛, 𝑥 ; 1) + 𝑝 ( 𝑛, 𝑥 ; 2) for 𝑛 ≥ , 𝑥 > . Introduce 𝐺 ( 𝑧, 𝑠 ) ∶= ∑ 𝑛 ≥ 𝑧 𝑛 ∫ ∞0 𝑒 − 𝑠𝑥 𝑔 ( 𝑛, 𝑥 ) 𝑑𝑥 ∀ | 𝑧 | ≤ , 𝑠 ≥ . Denote by 𝐹 ∗ 𝑍 ( 𝑠 ) = ∫ ∞0 𝑒 − 𝑠𝑥 𝑑𝐹 𝑍 ( 𝑥 ) the LST of 𝐹 𝑍 for 𝑠 ≥ . Note that ∫ ∞0 𝑒 − 𝑠𝑥 𝐹 𝑍 or 𝑋 ( 𝑥 ) 𝑑𝑥 = 𝐹 ∗ 𝑍 or 𝑋 ( 𝑠 ) 𝑠 , ∀ 𝑠 > . Multiplying both sides of (20) by 𝑧 𝑛 𝑒 − 𝑠𝑥 , integrating over 𝑥 ∈ [0 , ∞) and summing over all 𝑛 ≥ , yields 𝑠 ( 𝜆 (1 − 𝑧 ) − 𝑠 ) 𝐺 ( 𝑧, 𝑠 ) = 𝜆𝑧𝑝 (0) 𝐹 ∗ 𝑍 ( 𝑠 ) − ∑ 𝑛 ≥ 𝑧 𝑛 𝜕𝜕𝑥 𝑔 ( 𝑛,
0) + 𝐹 ∗ 𝑋 ( 𝑠 ) ∑ 𝑛 ≥ 𝑧 𝑛 𝜕𝜕𝑥 𝑔 ( 𝑛 + 𝑐, (21)where 𝜆𝑝 (0) = ∑ 𝑐𝑘 =1 𝜕𝜕𝑥 𝑔 ( 𝑘, from (19). We have 𝑧 𝑐 ∑ 𝑛 ≥ 𝑧 𝑛 + 𝑐 𝜕𝜕𝑥 𝑔 ( 𝑛 + 𝑐, 𝑧 𝑐 ∑ 𝑛 ≥ 𝑧 𝑛 𝜕𝜕𝑥 𝑔 ( 𝑛,
0) − 1 𝑧 𝑐 𝐻 ( 𝑧 ) (22)where 𝐻 ( 𝑧 ) = ∑ 𝑐𝑘 =1 𝑧 𝑘 𝑎 𝑘 with 𝑎 𝑘 ∶= 𝜕𝜕𝑥 𝑔 ( 𝑘, , for 𝑘 = 1 , … , 𝑐 . Introducing the above into (21) gives 𝑠 ( 𝜆 (1 − 𝑧 ) − 𝑠 ) 𝐺 ( 𝑧, 𝑠 ) = ( 𝐹 ∗ 𝑋 ( 𝑠 ) 𝑧 𝑐 − 1 ) Ψ( 𝑧 ) − 𝐹 ∗ 𝑋 ( 𝑠 ) 𝐻 ( 𝑧 ) 𝑧 𝑐 + 𝜆𝑧𝑝 (0) 𝐹 ∗ 𝑍 ( 𝑠 ) (23) where Ψ( 𝑧 ) ∶= ∑ 𝑛 ≥ 𝑧 𝑛 𝜕𝜕𝑥 𝑔 ( 𝑛, . Since 𝐺 ( 𝑧, 𝑠 ) is well-defined for | 𝑧 | ≤ and 𝑠 ≥ , the r.h.s. of (23) must vanishwhen 𝑠 = 𝜆 (1 − 𝑧 ) . This gives the relation Ψ( 𝑧 ) = 𝑧 𝑐 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) [ − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) 𝐻 ( 𝑧 ) 𝑧 𝑐 + 𝜆𝑧𝑝 (0) 𝐹 ∗ 𝑍 ( 𝜃 ( 𝑧 )) ] with 𝜃 ( 𝑧 ) = 𝜆 (1 − 𝑧 ) and | 𝑧 | ≤ . Introducing the above in (23) gives 𝑠 ( 𝜆 (1 − 𝑧 ) − 𝑠 ) 𝐺 ( 𝑧, 𝑠 ) = − 𝐹 ∗ 𝑋 ( 𝑠 ) 𝐻 ( 𝑧 ) 𝑧 𝑐 + 𝜆𝑧𝑝 (0) 𝐹 ∗ 𝑍 ( 𝑠 ) + 𝐹 ∗ 𝑋 ( 𝑠 ) − 𝑧 𝑐 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) [ 𝜆𝑧𝑝 (0) 𝐹 ∗ 𝑍 ( 𝜃 ( 𝑧 )) − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) 𝐻 ( 𝑧 ) 𝑧 𝑐 ] . (24) Panigrahy et al.:
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Let 𝑁 ( 𝑧 ) be the 𝑧 -transform of the stationary number of customers in the system. Integrating by part, we get for 𝑛 ≥ , 𝑠 ∫ ∞0 𝑒 − 𝑠𝑥 𝑔 ( 𝑛, 𝑥 ) 𝑑𝑥 = ∫ ∞0 𝑒 − 𝑠𝑥 𝑑𝑔 ( 𝑛, 𝑥 ) , so that lim 𝑠 → ∞ 𝑠 ∫ ∞0 𝑒 − 𝑠𝑥 𝑔 ( 𝑛, 𝑥 ) 𝑑𝑥 = lim 𝑠 → ∫ ∞0 𝑒 − 𝑠𝑥 𝑑𝑔 ( 𝑛, 𝑥 ) = ∫ ∞0 𝑑𝑔 ( 𝑛, 𝑥 ) = 𝑔 ( 𝑛, ∞) , (25)where the interchange between the limit and the integral sign is justified by the bounded convergence theorem. There-fore, 𝑁 ( 𝑧 ) = ∑ 𝑛 ≥ 𝑧 𝑛 𝑔 ( 𝑛, ∞) + 𝑝 (0)= ∑ 𝑛 ≥ 𝑧 𝑛 lim 𝑠 → ∞ 𝑠 ∫ ∞0 𝑒 − 𝑠𝑥 𝑔 ( 𝑛, 𝑥 ) 𝑑𝑥 from (25)= lim 𝑠 → 𝑠𝐺 ( 𝑧, 𝑠 ) + 𝑝 (0) , (26)where the interchange between the summation over 𝑛 and the integral sign is again justified by the bounded convergencetheorem. Letting now 𝑠 → in (24) and using (26), gives 𝜃 ( 𝑧 ) 𝑁 ( 𝑧 ) = 1 − 𝑧 𝑐 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) [ − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) 𝐻 ( 𝑧 ) 𝑧 𝑐 + 𝜆𝑧𝑝 (0) 𝐹 ∗ 𝑍 ( 𝜃 ( 𝑧 )) ] − 𝐻 ( 𝑧 ) 𝑧 𝑐 + 𝜆𝑝 (0) . (27)By noting that 𝜆𝑝 (0) = ∑ 𝑐𝑘 =1 𝑎 𝑘 (cf. (19)), Eq. (27) can be rewritten as 𝑁 ( 𝑧 ) = 1 𝜃 ( 𝑧 ) ( 𝑧 (1 − 𝑧 𝑐 ) 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) 𝑐 ∑ 𝑘 =1 𝑎 𝑘 [ 𝐹 ∗ 𝑍 ( 𝜃 ( 𝑧 )) − 𝑧 𝑘 − 𝑐 −1 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) ] + 𝑐 ∑ 𝑘 =1 𝑎 𝑘 (1 − 𝑧 𝑘 − 𝑐 ) ) . (28)The r.h.s. of (28) contains 𝑐 unknown constants 𝑎 , … , 𝑎 𝑐 yet to be determined. Define 𝐴 ( 𝑧 ) = 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) . It canbe shown that 𝑧 𝑐 − 𝐴 ( 𝑧 ) has 𝑐 − 1 zeros inside and one on the unit circle, | 𝑧 | = 1 (See Appendix 13.2.3). Denoteby 𝜉 , … , 𝜉 𝑞 the ≤ 𝑞 ≤ 𝑐 distinct zeros of 𝑧 𝑐 − 𝐴 ( 𝑧 ) in { | 𝑧 | ≤ , with multiplicity 𝑛 , … , 𝑛 𝑞 , respectively, with 𝑛 + ⋯ + 𝑛 𝑞 = 𝑐 . Hence, 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝑘 ( 𝑧 )) = 𝛾 𝑞 ∏ 𝑖 =1 ( 𝑧 − 𝜉 𝑖 ) 𝑛 𝑖 . Since 𝑧 𝑐 − 𝐴 ( 𝑧 ) vanishes when 𝑧 = 1 and that 𝑑𝑑𝑧 ( 𝑧 𝑐 − 𝐴 ( 𝑧 )) | 𝑧 =1 = 𝑐 − 𝜌 > , we conclude that 𝑧 𝑐 − 𝐴 ( 𝑧 ) has one zeroof multiplicity one at 𝑧 = 1 .Without loss of generality assume that 𝜉 𝑞 = 1 and let us now focus on the zeros 𝜉 , … , 𝜉 𝑞 −1 . When 𝑧 = 𝜉 𝑖 , 𝑖 = 1 , … , 𝑞 − 1 , the term 𝐹 ∗ 𝑍 ( 𝜃 ( 𝑧 )) − 𝑧 𝑘 − 𝑐 −1 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) in (28) must have a zero of multiplicity (at least) 𝑛 𝑖 since 𝑁 ( 𝜉 𝑖 ) is well defined. This gives 𝑐 − 1 linear equations to be satisfied by 𝜉 , … , 𝜉 𝑞 . In the particular case where all zeroshave multiplicity one (see Appendix 13.2.2), namely 𝑞 = 𝑐 , these 𝑐 − 1 equations are 𝑐 ∑ 𝑘 =1 𝑎 𝑘 [ 𝐹 ∗ 𝑍 ( 𝜃 ( 𝜉 𝑖 )) − 𝜉 𝑘 − 𝑐 −1 𝑖 𝐹 ∗ 𝑋 ( 𝜃 ( 𝜉 𝑖 )) ] = 0 , 𝑖 = 1 , … , 𝑐 − 1 . (29)With 𝑈 ( 𝑧 ) ∶= 𝐹 ∗ 𝑍 ( 𝜃 ( 𝑧 ))∕ 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) (29) is equivalent to 𝑐 ∑ 𝑘 =1 𝑎 𝑘 [ 𝑈 ( 𝜉 𝑖 ) − 𝜉 𝑘 − 𝑐 −1 𝑖 ) ] = 0 , 𝑖 = 1 , … , 𝑐 − 1 , (30) Panigrahy et al.:
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Page 11 of 28ne-dimensional Distributed Service Networks E xp ec t e d R e qu e s t D i s t a n ce r a ti o DeterministicUniform
Figure 3:
The plot shows the ratio 𝔼 [ 𝐷 ]∕ 𝐷 𝑠 for deterministic and uniform inter-server distance distributions. since 𝐹 ∗ 𝑋 ( 𝜃 ( 𝜉 𝑖 )) ≠ for 𝑖 = 1 , … , 𝑐 −1 ( 𝐹 ∗ 𝑋 ( 𝜃 ( 𝜉 𝑖 )) = 0 implies that 𝜉 𝑖 =0 which contradicts that 𝜉 𝑖 a zero of 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) since 𝐹 ∗ 𝑋 ( 𝜃 (0)) = 𝐹 ∗ 𝑋 ( 𝜆 ) > ). Eq. (28) can be rewritten as 𝑁 ( 𝑧 ) = ∑ 𝑐𝑘 =1 𝑎 𝑘 [ 𝑧 𝑐 − 𝑧 𝑘 + 𝑧 (1 − 𝑧 𝑐 ) 𝐹 ∗ 𝑍 ( 𝜃 ( 𝑧 )) − (1 − 𝑧 𝑘 ) 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) ] 𝜃 ( 𝑧 )( 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) . (31) A 𝑐 -th equation is provided by the normalizing condition 𝑁 ( 𝑧 ) = 1 . Since the numerator and denominator in (31)have a zero of order at 𝑧 = 1 , differentiating twice the numerator and the denominator w.r.t 𝑧 and letting 𝑧 = 1 gives 𝑐 ∑ 𝑘 =1 𝑎 𝑘 ( 𝑐 (1 + 𝜌 𝑧 ) − 𝜌𝑘 ) = 𝜆 ( 𝑐 − 𝜌 ) , (32)where 𝜌 𝑧 = 𝜆𝛼 𝑍 . We consider few special cases of the model in Appendix 13.2.4 and verify with the expressions ofqueue length distribution available in the literature.
From (31) the expected queue length is 𝑁 = 𝑑𝑑𝑧 𝑁 ( 𝑧 ) ||| 𝑧 =1 = 12 𝜆 ( 𝑐 − 𝜌 ) 𝑐 ∑ 𝑘 =1 𝑎 𝑘 [ 𝜆 𝜎 (2) 𝑍 𝑐 ( 𝑐 − 𝜌 ) + 𝜆 𝜎 (2) 𝑋 𝑐 (1 + 𝜌 𝑧 − 𝑘 ) + ( 𝑐𝑘 ( 𝑐 − 𝑘 ) + 𝑘 ( 𝑘 − 1) 𝜌 − 𝑐 ( 𝑐 − 1)) 𝜌 + 2 𝑐 𝜌 𝑧 − 𝑐 ( 𝑐 + 1) 𝜌 𝑧 𝜌 ] , (33) where 𝜎 (2) 𝑍 and 𝜎 (2) 𝑋 are the second order moments of distributions 𝐹 𝑍 and 𝐹 𝑋 respectively. Again by applying Little’slaw and considering the analogy between our service network with ESABQ we get the following expression for theexpected request distance. 𝔼 [ 𝐷 ] = 𝑁 ∕ 𝜆. (34)
7. Discussion of Unidirectional Allocation Policies
In this section we describe generalizations of models and results for unidirectional allocation policies. We firstconsider the case when inter-user and inter-server distances both have general distributions.
Consider the case when the inter-user and inter-server distances each are described by general distributions. Weassume server capacity, 𝑐 = 1 . As 𝜌 → , we conjecture that the behavior of MTR approaches that of the G/G/1queue. One argument in favor of our conjecture is the following. As 𝜌 → , the busy cycle duration tends to infinity.Consequently, the impact of the exceptional service for the first customer of the busy period on all other customersdiminishes to zero as there is an unbounded increasing number of customers served in the busy period. Panigrahy et al.:
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Page 12 of 28ne-dimensional Distributed Service Networks
It is known that in heavy traffic waiting times in a G/G/1 queue are exponential distributed and the mean sojourntime is given by 𝛼 𝑋 +[( 𝜎 𝑋 + 𝜎 𝑌 )∕2 𝛼 𝑌 (1− 𝜌 )] [20]. We expect the expected request distance to exhibit similar behavior.Thus we have the following conjecture. Conjecture 1.
At heavy traffic i.e. as 𝜌 → , the expected request distance for the G/G/1 spatial system with 𝑐 = 1 isgiven by 𝔼 [ 𝐷 ] = 𝛼 𝑋 + 𝜎 𝑋 + 𝜎 𝑌 𝛼 𝑌 (1 − 𝜌 ) . (35)Denote by 𝐷 𝑠 the average request distance as obtained from simulation. We plot the ratio 𝔼 [ 𝐷 ]∕ 𝐷 𝑠 across variousinter-request and inter-server distance distributions in Figure 3. It is evident that as 𝜌 → , the ratio 𝔼 [ 𝐷 ]∕ 𝐷 𝑠 convergesto across different inter-server distance distributions. We now proceed to analyze a setting where server capacity is a random variable. Assume server capacity takesvalues from {1 , , … , 𝑐 } with distribution Pr ( = 𝑗 ) = 𝑝 𝑗 , ∀ 𝑗 ∈ {1 , , … , 𝑐 } , s.t. ∑ 𝑐𝑗 =1 𝑝 𝑗 = 1 and 𝑝 𝑐 > . Wealso assume the stability condition 𝜌 < where is the average server capacity. Denote 𝐻 as the random variableassociated with number of requests that traverse through a point just after a server location . 𝐻 Let 𝑉 denote the number of new requests generated during a service period with 𝑘 𝑣 = Pr ( 𝑉 = 𝑣 ) , ∀ 𝑣 ≥ . According to the law of total probability, it holds that 𝑘 𝑣 = ∞ ∫ Pr ( 𝑉 = 𝑣 | 𝑋 = 𝜈 ) 𝑓 𝑋 ( 𝜈 ) = 1 𝑣 ! ∞ ∫ 𝑒 − 𝜆𝜈 ( 𝜆𝜈 ) 𝑣 𝑑𝐹 𝑋 ( 𝜈 ) . (36)Then the corresponding generating function 𝐾 ( 𝑧 ) is denoted by 𝐾 ( 𝑧 ) = ∞ ∑ 𝑣 =0 𝑘 𝑣 𝑧 𝑣 = 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) . (37)We now consider an embedded Markov chain generated by 𝐻 . Denote the corresponding transition matrix as 𝑀. Thenwe have 𝑀 𝑚,𝑙 = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ 𝑐 − 𝑚 ∑ 𝑖 =0 𝑘 𝑖 𝑃 𝑖 + 𝑚 , ≤ 𝑚 ≤ 𝑐, 𝑙 = 0; 𝑐 ∑ 𝑖 =0 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 , ≤ 𝑚 ≤ 𝑙, 𝑙 ≠ 𝑐 ∑ 𝑖 = 𝑚 − 𝑙 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 , 𝑙 + 1 ≤ 𝑚 ≤ 𝑐 + 𝑙, 𝑙 ≠ , 𝑜.𝑤., (38) where 𝑃 𝑖 = ∑ 𝑐𝑗 = 𝑖 𝑝 𝑗 and 𝑝 = 0 . Let 𝜋 = [ 𝜋 𝑗 , 𝑗 ≥ and 𝑁 ( 𝑧 ) = ∑ 𝑗 ≥ 𝜋 𝑗 𝑧 𝑗 denote the steady state distribution andits 𝑧 -transform respectively. 𝜋 is obtained out by solving 𝜋 𝑙 = ∞ ∑ 𝑚 =0 𝜋 𝑚 𝑀 𝑚,𝑙 , 𝑙 = 0 , , … . (39)Thus we have for 𝑙 ∈ ℕ ,𝜋 = 𝑐 ∑ 𝑚 =0 𝜋 𝑚 𝑐 − 𝑚 ∑ 𝑖 =0 𝑘 𝑖 𝑃 𝑖 + 𝑚 , 𝜋 𝑙 = 𝑙 ∑ 𝑚 =0 𝜋 𝑚 𝑐 ∑ 𝑖 =0 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 + 𝑐 + 𝑙 ∑ 𝑚 = 𝑙 +1 𝜋 𝑚 𝑐 ∑ 𝑖 = 𝑚 − 𝑙 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 . (40) An analysis for the distribution of number of requests that traverse through any random location would involve the notions of exceptionalservice and accessible batches.
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Page 13 of 28ne-dimensional Distributed Service Networks
Multiplying by 𝑧 𝑙 and summing over 𝑙 gives 𝑁 ( 𝑧 ) = 𝐸 𝜋 + 𝑣 ( 𝑧 ) + 𝑣 ( 𝑧 ) (41) 𝐸 𝜋 = 𝜋 𝑐 −1 ∑ 𝑖 =0 𝑘 𝑖 𝑃 𝑖 +1 + 𝑐 −1 ∑ 𝑚 =1 𝜋 𝑚 𝑐 −1 ∑ 𝑖 = 𝑚 𝑘 𝑖 − 𝑚 𝑃 𝑖 +1 (42) 𝑣 ( 𝑧 ) = ∞ ∑ 𝑙 =0 𝑧 𝑙 𝑙 ∑ 𝑚 =0 𝜋 𝑚 𝑐 ∑ 𝑖 =0 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 (43) 𝑣 ( 𝑧 ) = ∞ ∑ 𝑙 =0 𝑧 𝑙 𝑐 + 𝑙 ∑ 𝑚 = 𝑙 +1 𝜋 𝑚 𝑐 ∑ 𝑖 = 𝑚 − 𝑙 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 . (44) The expressions for 𝑣 ( 𝑧 ) and 𝑣 ( 𝑧 ) can be further simplified (see Appendix 13.3) to 𝑣 ( 𝑧 ) = 𝑁 ( 𝑧 ) { 𝑐 ∑ 𝑖 =0 𝑝 𝑖 𝑧 − 𝑖 [ 𝐾 ( 𝑧 ) − 𝑖 ∑ 𝑗 =0 𝑘 𝑗 𝑧 𝑗 ] + 𝑐 ∑ 𝑖 =0 𝑘 𝑖 𝑧 𝑖 } (45) 𝑣 ( 𝑧 ) = [ 𝑐 ∑ 𝑚 =0 𝑧 − 𝑚 𝑐 ∑ 𝑖 = 𝑚 𝑘 𝑖 − 𝑚 𝑝 𝑖 { 𝑁 ( 𝑧 ) − 𝑚 −1 ∑ 𝑗 =0 𝜋 𝑗 𝑧 𝑗 }] − 𝑁 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑘 𝑖 𝑧 𝑖 . (46) Combining (41), (45) and (46) yields 𝑁 ( 𝑧 ) = 𝐸 𝜋 + 𝑁 ( 𝑧 ) { 𝐾 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑝 𝑖 𝑧 − 𝑖 } − 𝑐 −1 ∑ 𝑗 =0 𝜋 𝑗 𝑐 − 𝑗 ∑ 𝑚 =1 𝑧 − 𝑚 𝑐 ∑ 𝑖 = 𝑚 + 𝑗 𝑘 𝑖 −( 𝑚 + 𝑗 ) 𝑝 𝑖 . (47) Thus we obtain 𝑁 ( 𝑧 ) = 𝐸 𝜋 − 𝑐 −1 ∑ 𝑗 =0 𝜋 𝑗 𝑐 − 𝑗 ∑ 𝑚 =1 𝑧 − 𝑚 𝑐 ∑ 𝑖 = 𝑚 + 𝑗 𝑘 𝑖 −( 𝑚 + 𝑗 ) 𝑝 𝑖 𝐾 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑝 𝑖 𝑧 − 𝑖 . (48) Multipying numerator and denominator by 𝑧 𝑐 yields 𝑁 ( 𝑧 ) = 𝑧 𝑐 𝐸 𝜋 − 𝑐 −1 ∑ 𝑗 =0 𝜋 𝑗 𝑐 − 𝑗 ∑ 𝑚 =1 𝑧 𝑐 − 𝑚 𝑐 ∑ 𝑖 = 𝑚 + 𝑗 𝑘 𝑖 −( 𝑚 + 𝑗 ) 𝑝 𝑖 𝑧 𝑐 − 𝐾 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑝 𝑐 − 𝑖 𝑧 𝑖 . (49) To determine 𝑁 ( 𝑧 ) , we need to obtain the probabilities 𝜋 𝑖 , ≤ 𝑖 ≤ 𝑐 − 1 . It can be shown that the denominator of (49)has 𝑐 − 1 zeros inside and one on the unit circle, | 𝑧 | = 1 (See Appendix 13.3.2). As 𝑁 ( 𝑧 ) is analytic within and on theunit circle, the numerator must vanish at these zeros, giving rise to 𝑐 equations in 𝑐 unknowns.Let 𝜉 𝑞 ∶ 1 ≤ 𝑞 ≤ 𝑐 be the zeros of 𝑧 𝑐 − 𝐾 ( 𝑧 ) ∑ 𝑐𝑖 =0 𝑝 𝑐 − 𝑖 𝑧 𝑖 in { | 𝑧 | ≤ . W.l.o.g let 𝜉 𝑐 = 1 . We have the following 𝑐 − 1 equations. 𝐸 𝜋 − 𝑐 −1 ∑ 𝑗 =0 𝜋 𝑗 𝑐 − 𝑗 ∑ 𝑚 =1 𝜉 − 𝑚𝑞 𝑐 ∑ 𝑖 = 𝑚 + 𝑗 𝑘 𝑖 −( 𝑚 + 𝑗 ) 𝑝 𝑖 = 0 , 𝑖 = 1 , … , 𝑐 − 1 , (50) A 𝑐 -th equation is provided by the normalizing condition lim 𝑧 → 𝑁 ( 𝑧 ) = 1 . In the particular case where allzeros have multiplicity one, it can be shown that these 𝑐 equations are linearly independent . Once the parameters { 𝜋 𝑖 , ≤ 𝑖 ≤ 𝑐 − 1} are known, 𝔼 [ 𝐻 ] can be expressed as 𝔼 [ 𝐻 ] = 𝐻 = lim 𝑧 → 𝑁 ′ ( 𝑧 ) . (51) To evaluate the expected request distance we adopt arguments from [6]. Consider any interval of length 𝜈 betweentwo consecutive servers. There are on average 𝐻 requests at the beginning of the interval , each of which must travel 𝜈 distance. New users are spread randomly over the interval and there are on an average 𝜆𝜈 new users. The requestmade by each new user must travel on average 𝜈 ∕2 . Thus we have 𝔼 [ 𝐷 ] = 1 𝜌 ∫ ∞0 ( 𝐻𝜈 + 12 𝜆𝜈 ) 𝑑𝐹 𝑋 ( 𝜈 ) = 1 𝜌 [ 𝐻𝜇 + 𝜆 ( 𝜎 𝑋 + 1 𝜇 )] . (52) For all cases evaluated across uniform, deterministic and hyperexponential distributions we found the set of 𝑐 equations to be linearly inde-pendent. Panigrahy et al.:
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Page 14 of 28ne-dimensional Distributed Service Networks
An interesting special case of the unidirectional general matching is the uncapacitated scenario. Consider the casewhere servers do not have any capacity constraints, i.e. 𝑐 = ∞ . In such a case, all users are assigned to the nearestserver to their right.
GRPS:
When 𝑐 → ∞ and given < 𝑟 < , 𝑟 = 𝐹 ∗ 𝑌 ( 𝜇 − 𝜇𝑟 𝑐 ) = 𝐹 ∗ 𝑌 ( 𝜇 ) . Setting 𝜔 = 1∕ 𝐹 ∗ 𝑌 ( 𝜇 ) = 1∕ 𝑟 in (16) andsimplifying yields 𝐶 → , as 𝑐 → ∞ , ⟹ 𝔼 [ 𝐷 ] → 𝜇 as 𝑐 → ∞ . (53) PRGS:
Under PRGS, when 𝑐 → ∞ there exists no request allocated to a server other than the nearest server to itsright. Again using Bailey’s method as in [6] and setting 𝐻 = 0 in (52) we get 𝔼 [ 𝐷 ] → 𝜇 ( 𝜎 𝑋 + 1 𝜇 ) as 𝑐 → ∞ . (54) Consider the following generalization of the service network. We define cost of an allocation as the communicationcost associated with an allocated request-server pair. Consider communication cost as a function of the requestdistances. Then the expected communication cost across the service network is given as 𝑇 = 𝐸 [ cost ] = ∞ ∫ 𝑑 =0 ( 𝑑 ) 𝑑𝑊 ( 𝑑 ) , (55)where 𝑊 is the request distance distribution. One such cost model widely used in wireless ad hoc networks is [8] ( 𝑑 ) = 𝑡 𝑑 𝛽 , (56)where 𝛽 is the path loss exponent typically ≤ 𝛽 ≤ and 𝑡 is a constant. Below we derive the expected communi-cation cost for the scenario when 𝑐 = 1 . 𝑐 = 1 In this case the service network directly maps to a temporal G/M/1 queue. Thus 𝑊 can be expressed as the sojourntime distribution of the corresponding G/M/1 queue. Hence 𝑊 ∼ Expo ( 𝜇 (1 − 𝑟 )) with 𝑟 as defined in Section 6.2.We have 𝑇 = ∞ ∫ 𝑑 =0 𝑡 𝑑 𝛽 𝑑𝑊 ( 𝑑 ) = 𝑡 𝜇 𝛽 (1 − 𝑟 ) 𝛽 Γ( 𝛽 + 1) , (57)where Γ( 𝑥 ) = ∫ ∞0 𝑦 𝑥 −1 𝑒 − 𝑦 𝑑𝑦 is the gamma function. 𝑐 = 1 In this case, the service network can be modeled as a temporal M/G/1 queue with first customer having exceptionalservice [22]. Denote 𝑊 ∗ ( 𝑠 ) as the LS transform of 𝑊 .
Using results from [22] 𝑊 ∗ ( 𝑠 ) = (1 − 𝜌 ) { 𝜆 [ 𝐹 ∗ 𝑍 ( 𝑠 ) − 𝐹 ∗ 𝑋 ( 𝑠 ) ] − 𝑠𝐹 ∗ 𝑍 ( 𝑠 ) } (1 − 𝜌 + 𝜌 𝑍 ) [ 𝜆 − 𝑠 − 𝜆𝐹 ∗ 𝑋 ( 𝑠 ) ] . (58)When 𝛽 is an integer, 𝑇 = 𝑡 (−1) 𝛽 𝑑 ( 𝛽 ) 𝑑𝑠 𝑊 ∗ ( 𝑠 ) | 𝑠 =0 , (59) Panigrahy et al.:
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Page 15 of 28ne-dimensional Distributed Service Networks (a) (b)
Figure 4:
Two resource scenario with 𝑐 = 1 (a) Depiction of request distances and (b) Mapping to Fork-join queues. Now consider the following scenario where each user requests two resources which reside on different servers asshown in Figure 4(a). Let the corresponding servers be distributed according to a Poisson process with densities 𝜇 and 𝜇 . Let the users be distributed according to a Poisson process. The service network, in this case, can be modeled as afork-join queueing system as shown in Figure 4(b) [16]. In such a queue, each incoming job is split into two sub-jobseach of which is served on one of the two servers. After service, each sub-job waits until the other sub-job has beenprocessed. They then merge and leave the system. In the service network as well, each request forks two sub-requestsone for each resource type. A request is said to be completed only if it has retrieved both the resources, thus mappingit to a fork-join queue. We define the overall request distance to be the maximum value among the request distancesacross all resource types and denote it as the random variable 𝐷 𝑚𝑎𝑥 . 𝜇 = 𝜇 = 𝜇𝜇 = 𝜇 = 𝜇𝜇 = 𝜇 = 𝜇 and 𝑐 = 1 𝑐 = 1 𝑐 = 1 ) The approximated expected request distance for this scenario is obtained from the expression for the expectedsojourn time of a fork join queue with homogeneous servers as [16]: 𝔼 [ 𝐷 𝑚𝑎𝑥 ] = 12 𝜇 − 𝜆 𝜇 ( 𝜇 − 𝜆 ) , (60)Note that, the corresponding expected request distance in case of single resource is given by Equation (6) 𝔼 [ 𝐷 ] =1∕( 𝜇 − 𝜆 ) . Clearly, 𝔼 [ 𝐷 𝑚𝑎𝑥 ] = 12 𝜇 − 𝜆 𝜇 ( 𝜇 − 𝜆 ) = [1 . . 𝜌 ] 1 𝜇 − 𝜆 > 𝜇 − 𝜆 = 𝔼 [ 𝐷 ] , (61)Thus we have 𝔼 [ 𝐷 𝑚𝑎𝑥 ] > 𝔼 [ 𝐷 ] .
8. Bidirectional Allocation Policies
Both UGS and MTR minimize expected request distance among all unidirectional policies. In this section weformulate the bi-directional allocation policy that minimizes expected request distance. Let 𝜂 ∶ 𝑅 → 𝑆 be anymapping of users to servers. Our objective is to find a mapping 𝜂 ∗ ∶ 𝑅 → 𝑆 , that satisfies 𝜂 ∗ = arg min 𝜂 ∑ 𝑖 ∈ 𝑅 𝑑 ( 𝑟 𝑖 , 𝑠 𝜂 ( 𝑖 ) ) 𝑠.𝑡. ∑ 𝑖 ∈ 𝑅 𝜂 ( 𝑖 )= 𝑗 ≤ 𝑐, ∀ 𝑗 ∈ 𝑆 (62)W.l.o.g, let 𝑟 ≤ 𝑟 ≤ ⋯ ≤ 𝑟 𝑖 ≤ ⋯ ≤ 𝑟 | 𝑅 | be locations of requests and 𝑠 ≤ 𝑠 ≤ ⋯ ≤ 𝑠 𝑖 ≤ ⋯ ≤ 𝑠 | 𝑆 | be locations ofservers. We first focus on the case when 𝑐 = 1 . We consider the following two scenarios. Case 1: | 𝑅 | = | 𝑆 || 𝑅 | = | 𝑆 || 𝑅 | = | 𝑆 | When | 𝑅 | = | 𝑆 | , an optimal allocation strategy is given by the following theorem [7]. Theorem 2.
When | 𝑅 | = | 𝑆 | , an optimal assignment is obtained by the policy: 𝜂 ∗ ( 𝑖 ) = 𝑖, ∀ 𝑖 ∈ {1 , ⋯ , | 𝑅 | } i.e.allocating the 𝑖 𝑡ℎ request to the 𝑖 𝑡ℎ server and the average request distance is given by 𝔼 [ 𝐷 ] = 1 | 𝑅 | | 𝑅 | ∑ 𝑖 =1 | 𝑠 ( 𝑖 ) − 𝑟 ( 𝑖 ) | . (63) Panigrahy et al.:
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Page 16 of 28ne-dimensional Distributed Service Networks ℇ ℇ ℇ ℇ ℇ ℇ ℇ ℇ Optimal minimum weight matchingGale-Shapley stable matching
Figure 5:
Worst case scenario for Gale-Shapley.
Case 2: | 𝑅 | < | 𝑆 || 𝑅 | < | 𝑆 || 𝑅 | < | 𝑆 | This is the case where there are fewer requesters than servers. In this case, a Dynamic Programming( DP ) based algorithm (Algorithm 1) obtains the optimal assignment.Let 𝐶 [ 𝑖, 𝑗 ] denote the optimal cost (i.e., sum of distances) of assigning the first 𝑖 requests (counting from the left)located at 𝑟 ≤ 𝑟 ≤ … ≤ 𝑟 𝑖 to the first 𝑗 servers (also counting from the left) located at 𝑠 ≤ 𝑠 ≤ … ≤ 𝑠 𝑗 . If 𝑗 == 𝑖 ,the optimal assignment is trivial due to Theorem 2 and 𝐶 [ 𝑖, 𝑖 ] is computed easily for all 𝑖 ≤ | 𝑅 | by summing pairwisedistances 𝑑 [1 , , 𝑑 [2 , , … , 𝑑 [ 𝑖, 𝑖 ] (Lines 6–7). For the base case, 𝑖 = 1 , 𝑗 > , only the first user needs to be assignedto its nearest server (Lines 9–16). For the general dynamic programming step, consider 𝑗 > 𝑖 . Then 𝐶 [ 𝑖, 𝑗 ] can beexpressed in terms of the costs of two subproblems, i.e., 𝐶 [ 𝑖 − 1 , 𝑗 − 1] and 𝐶 [ 𝑖, 𝑗 − 1] (Lines 19–24). In the optimalsolution, two cases are possible: either request 𝑖 is assigned to server 𝑗 , or the latter is left unallocated. The formercase occurs if the first 𝑖 − 1 requests are assigned to the first 𝑗 − 1 servers at cost 𝐶 [ 𝑖 − 1 , 𝑗 − 1] , and the latter caseoccurs when the first 𝑖 requests are assigned to the first 𝑗 − 1 servers at cost 𝐶 [ 𝑖, 𝑗 − 1] . This is a consequence of theno-crossing lemma (Lemma 1). The optimal 𝐶 [ 𝑖, 𝑗 ] is chosen depending on these two costs and the current distance 𝑑 [ 𝑖, 𝑗 ] . Lemma 1.
In an optimal solution, 𝜂 ∗ , to the problem of matching users at 𝑟 ≤ 𝑟 ≤ … ≤ 𝑟 | 𝑅 | to servers at 𝑠 ≤ 𝑠 ≤ … ≤ 𝑠 | 𝑆 | , where | 𝑆 | ≥ | 𝑅 | , there do not exist indices 𝑖, 𝑗 such that 𝜂 ∗ ( 𝑖 ) > 𝜂 ∗ ( 𝑖 ′ ) when 𝑖 ′ > 𝑖 .Proof. See Appendix 13.4.The dynamic programming algorithm fills cells in an | 𝑅 | × | 𝑆 | matrix 𝐶 whose origin is in the north-west corner.The lower triangular portion of this matrix is invalid since | 𝑅 | ≤ | 𝑆 | . The base cases populate the diagonal and thenorthernmost row, and in the general DP step, the value of a cell depends on the previously computed values in thecells located to its immediate west and diagonally north-west. As an optimization, for a fixed 𝑖 , the 𝑗 -th loop indexneeds to run only from 𝑖 + 1 through 𝑖 + | 𝑆 | − | 𝑅 | (Lines 11 and 18) instead of from 𝑖 + 1 through | 𝑆 | . This is becausethe first request has to be assigned to a server 𝑠 𝑗 with 𝑗 ≤ | 𝑆 | − | 𝑅 | + 1 so that the rest of the | 𝑅 | − 1 requests have achance of being placed on unique servers . The optimal average request distance is given by 𝐶 [ | 𝑅 | , | 𝑆 | ] .The time complexity of the main DP step is 𝑂 ( | 𝑅 | ×( | 𝑆 | − | 𝑅 | +1)) . Note that this assumes that the pairwise distancematrix 𝑑 of dimension | 𝑅 | × | 𝑆 | has been precomputed. The optimization applied above can be similarly applied tothis computation and hence the overall time complexity of Algorithm 1 is 𝑂 ( | 𝑅 | × ( | 𝑆 | − | 𝑅 | + 1)) . Therefore, if | 𝑆 | = 𝑂 ( | 𝑅 | ) , the worst case time complexity is quadratic in | 𝑅 | . However, if | 𝑆 | − | 𝑅 | grows only sub-linearly with | 𝑅 | , the time complexity is sub-quadratic in | 𝑅 | .Note that retrieving the optimal assignment requires more book-keeping. An | 𝑅 | × | 𝑆 | matrix 𝐴 stores key inter-mediate steps in the assignment as the DP algorithm progresses (Lines 8, 16, 21, 24). The optimal assignment vector 𝜋 can be retrieved from matrix 𝐴 using procedure R EAD O PT A SSIGNMENT .Another bidirectional assignment scheme is the Gale-Shapley algorithm [9], which produces stable assignments,though in the worst case it can yield an assignment that is 𝑂 ( | 𝑅 | ln 3∕2 ) ≈ 𝑂 ( | 𝑅 | . ) times costlier than the optimalassignment yielded by Algorithm 1, where | 𝑅 | is the number of users [18]. The worst case scenario is illustrated inFigure 5, with | 𝑅 | = 2 𝑡 −1 , where 𝑡 is the number of clusters of users and servers; and the largest distance betweenadjacent points is 𝑡 −2 . However at low/moderate loads for the cases evaluated in Section 9, we find its performanceto be not much worse than optimal. Note that in this exposition, we consider server capacity 𝑐 = 1 . If 𝑐 > , we simply add 𝑐 servers at each prescribed server location, andrequests will still be placed on unique servers. Panigrahy et al.:
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Page 17 of 28ne-dimensional Distributed Service Networks
Algorithm 1
Optimal Assignment by Dynamic Programming Input : 𝑟 ≤ ⋯ ≤ 𝑟 | 𝑅 | ; 𝑠 ≤ ⋯ ≤ 𝑠 | 𝑆 | Output : The optimal assignment 𝜋 procedure O PT DP( 𝑟, 𝑠 ) 𝑑 | 𝑅 | × | 𝑆 | = C OMPUTE P AIRWISE D ISTANCES ( 𝑟, 𝑠 ) 𝐶 = {∞} | 𝑅 | × | 𝑆 | for 𝑖 = 1 , ⋯ , | 𝑅 | do 𝐶 [ 𝑖, 𝑖 ] = T RIVIAL A SSIGNMENT ( 𝑖, 𝑑 ) 𝐴 [ | 𝑅 | , | 𝑅 | ] = | 𝑅 | 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 = 0 𝑛𝑒𝑎𝑟𝑒𝑠𝑡𝑐𝑜𝑠𝑡 = 𝐶 [1 , for 𝑗 = 2 , ⋯ , | 𝑆 | − | 𝑅 | + 1 do if 𝑑 [1 , 𝑗 ] < 𝑛𝑒𝑎𝑟𝑒𝑠𝑡𝑐𝑜𝑠𝑡 then 𝑛𝑒𝑎𝑟𝑒𝑠𝑡𝑐𝑜𝑠𝑡 = 𝑑 [1 , 𝑗 ] 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 = 𝑗 𝐶 [1 , 𝑗 ] = 𝑛𝑒𝑎𝑟𝑒𝑠𝑡𝑐𝑜𝑠𝑡 𝐴 [1 , 𝑗 ] = 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 for 𝑖 = 2 , ⋯ , | 𝑅 | do for 𝑗 = 𝑖 + 1 , ⋯ , 𝑖 + | 𝑆 | − | 𝑅 | do if 𝐶 [ 𝑖, 𝑗 − 1] < 𝑑 [ 𝑖, 𝑗 ] + 𝐶 [ 𝑖 − 1 , 𝑗 − 1] then 𝐶 [ 𝑖, 𝑗 ] = 𝐶 [ 𝑖, 𝑗 − 1] 𝐴 [ 𝑖, 𝑗 ] = 𝐴 [ 𝑖, 𝑗 − 1] else 𝐶 [ 𝑖, 𝑗 ] = 𝑑 [ 𝑖, 𝑗 ] + 𝐶 [ 𝑖 − 1 , 𝑗 − 1] 𝐴 [ 𝑖, 𝑗 ] = 𝑗 return R EAD O PT A SSIGNMENT ( 𝐴 ) procedure T RIVIAL A SSIGNMENT ( 𝑛, 𝑑 ) 𝐶𝑜𝑠𝑡 = 0 for 𝑖 = 1 , ⋯ , 𝑛 do 𝐶𝑜𝑠𝑡 = 𝐶𝑜𝑠𝑡 + 𝑑 [ 𝑖, 𝑖 ] return 𝐶𝑜𝑠𝑡 procedure R EAD O PT A SSIGNMENT ( 𝐴 ) | 𝑅 | , | 𝑆 | = D IMENSIONS ( 𝐴 ) 𝑠 = | 𝑆 | for 𝑖 = | 𝑅 | , ⋯ , do 𝜋 [ 𝑖 ] = 𝐴 [ 𝑖, 𝑠 ] 𝑠 = 𝐴 [ 𝑖, 𝑠 ] − 1 return 𝜋
9. Numerical Experiments
In this section, we examine the effect of various system parameters on expected request distance under MTR policy.We also compare the performance of various greedy allocation strategies along with the unidirectional policies to theoptimal strategy.
In our experiments, we consider a mean requester rate 𝜆 ∈ (0 , . We consider various inter-server distance distri-butions with density one. In particular, (i) for exponential distributions, the density is set to 𝜇 = 1 ; (ii) for deterministicdistributions, we assign parameter 𝑑 = 1 . (iii) for second order hyper-exponential distribution ( 𝐻 ), denote 𝑝 and 𝑝 as the phase probabilities. Let 𝜇 and 𝜇 be corresponding phase rates. We assume 𝑝 ∕ 𝜇 = 𝑝 ∕ 𝜇 . We express 𝐻 parameters in terms of the squared coefficient of variation, 𝑐 𝑣 , and mean inter-server distance, 𝛼 𝑋 , i.e. we set Panigrahy et al.:
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Page 18 of 28ne-dimensional Distributed Service Networks
Load E xp ec t e d R e qu e s t D i s t a n ce DeterministicExpoHyper-expo (a)
Squared Co-efficient of variance E xp ec t e d R e qu e s t D i s t a n ce M/H-2H-2/M (b)
Server Capacity E xp ec t e d R e qu e s t D i s t a n ce DeterministicExpoHyper-expo (c)
Server capacity E xp ec t e d R e qu e s t D i s t a n ce variable server capacityConstant server capacity (d) Load V a r i a n ce f o r R e qu e s t D i s t a n ce UGS (Deterministic)MTR (Deterministic)UGS (Expo)MTR (Expo)UGS (Hyper-expo)MTR (Hyper-expo) (e)
Deterministic Expo Hyper-expo
Distributions E x pe c t ed R eque s t D i s t an c e TRN ( , [0.9 , 1.1 ], c=1)TRH ( , [ , ], c=1)SRU ( , , c=1)SRB ( , /2, c=2) (f)
Figure 6:
Sensitivity analysis of MTR/UGS policy. (a) Effect of load on expected request distance with 𝑐 = 2 𝑐 = 2 𝑐 = 2 . (b)Effect of squared coefficient of variation on expected request distance with 𝜆 = 𝜇 = 1 𝜆 = 𝜇 = 1 𝜆 = 𝜇 = 1 and 𝑐 = 2 𝑐 = 2 𝑐 = 2 . (c) Effect of servercapacity on expected request distance with 𝜌 = 0 . 𝜌 = 0 . 𝜌 = 0 . . Effect of variability in server capacity on expected request distancefor (d) Deterministic distribution with 𝜌 = 0 . 𝜌 = 0 . 𝜌 = 0 . (e) Effect of load on variance of request distance with 𝑐 = 2 𝑐 = 2 𝑐 = 2 across MTRand UGS. (f) Comparison of expected request distance under Two Resource Non-homogeneous (TRN), Two ResourceHomogeneous (TRH), Single Resource Unit-service (SRU) and Single Resource Bulk-service (SRB) scenario across variousserver distributions with 𝜆 = 0 . , 𝜇 = 1 𝜆 = 0 . , 𝜇 = 1 𝜆 = 0 . , 𝜇 = 1 . 𝑝 = (1∕2) ( √ ( 𝑐 𝑣 − 1)∕( 𝑐 𝑣 + 1) ) , 𝑝 = 1 − 𝑝 , 𝜇 = 2 𝑝 ∕ 𝛼 𝑋 and 𝜇 = 2 𝑝 ∕ 𝛼 𝑋 . Unless specified, for 𝐻 we take 𝑐 𝑣 = 4 with 𝑐 = 2 . Also if not specified, users are distributed according to a Poisson process and servers a accordingto a renewal process.We consider a collection of users and servers, i.e. | 𝑅 | = | 𝑆 | = 10 . We assign users to servers accordingto MTR. Let 𝑅 𝑀 ⊆ 𝑅 be the set of users allocated under MTR. Clearly | 𝑅 𝑀 | ≤ | 𝑅 | . We then run optimal and othergreedy policies on the set 𝑅 𝑀 and 𝑆. For each of the experiments, the expected request distance for the correspondingpolicy is averaged over trials. We first study the effect of load ( = 𝜆 ∕ 𝑐𝜇 ) on 𝔼 [ 𝐷 ] as shown in Figure 6(a). Clearly 𝔼 [ 𝐷 ] increases as a functionof load. Note that 𝐻 distribution exhibits the largest expected request distance and the deterministic distribution,the smallest because the servers are evenly spaced. While for 𝐻 , 𝑐 𝑣 is larger than for the exponential distribution.Consequently servers are clustered, which increases 𝔼 [ 𝐷 ] . We now examine how 𝑐 𝑣 affects 𝔼 [ 𝐷 ] when 𝜌 is fixed. We compare two systems: a general request with Poissondistributed servers ( 𝐻 /M) and a Poisson request with general distributed servers (M/ 𝐻 ) where the general distributionis a 𝐻 distribution with the same set of parameters, i.e. we fix 𝜆 = 𝜇 = 1 with 𝑐 = 2 . The results are shown in Figure6(b). Note that, when 𝑐 𝑣 = 1 𝐻 is an exponential distribution and both 𝐻 /M and M/ 𝐻 are identical M/M/1 systems. Panigrahy et al.:
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Page 19 of 28ne-dimensional Distributed Service Networks
As discussed in the previous graph, performance of both systems decreases with increase in 𝑐 𝑣 due to increase in thevariability of user and server placements. However, from Figure 6(b) it is clear that performance is more sensitive toserver placement as compared to the corresponding user placement. We now focus on how server capacity affects 𝔼 [ 𝐷 ] as shown in Figure 6(c). We fix 𝜌 = 0 . . With an increase in 𝑐 ,while keeping 𝜌 fixed, 𝔼 [ 𝐷 ] decreases. This is because queuing delay decreases. Note that 𝔼 [ 𝐷 ] gradually convergeto a value with increase in server capacity. Theoretically, this can be explained by our discussion on uncapacitatedallocation in Section 7.3. As 𝑐 → ∞ the contribution of queuing delay to 𝔼 [ 𝐷 ] vanishes and 𝔼 [ 𝐷 ] becomes insensitiveto 𝑐. We investigate the heterogeneous capacity scenario as discussed in Section 7.2. Consider the plot shown in Figure6(d). We fix 𝜌 = 0 . . For the variable server capacity curve we choose a value for server capacity for each serveruniformly at random from the set {1 , , … , 𝑐 } . For the constant server capacity curve we deterministically assignserver capacity 𝑐 to each server. We observe better performance for constant server capacity curve at lower values of 𝑐 under Deterministic distribution. Variability in constant server case is zero, thus explaining its better performance.Both the curves exhibit similar performance under 𝐻 distribution as well. We now study the effect of load on variance of request distance as shown in Figure 6(e). Clearly variance increasesas a function of load. Also note that UGS has a higher variance as compared to MTR across all values of load andacross various inter-server distance distributions. Provable results exist (from queueing theory) that among all servicedisciplines the variance of the request distance (or sojourn time in queueing terminology) is minimized under MTR (aFCFS based policy) for Poisson request arrivals and exponential inter-server distances (or service times) [13]. However,these results do not generalize to other inter-server distance distributions in an exceptional service accessible batchqueueing discipline. Our simulation based results in Figure 6(e) thus bolster our observation in Remark 2 mentionedin Section 4. Again, a deterministic equidistant placement of servers produce the least variance for request distanceamong all other placements.
We compare the performance of MTR under various two resource (TR) and single resource (SR) settings as shownin Figure 6(f). For a two resource setting, denote [ 𝜇 , 𝜇 ] as the server densities associated with resource types and respectively as described in Section 7.5. Denote 𝑐 as the server capacity associated with each resource type. Wedefine a Two Resource Homogeneous (TRH) system to be a two resource setting with 𝜇 = 𝜇 = 𝜇 . We define aTwo Resource Non-homogeneous (TRN) system to be a two resource setting with 𝜇 ≠ 𝜇 . For simulation purpose,we chose 𝜇 = 𝜇 + 𝜖 and 𝜇 = 𝜇 − 𝜖 such that the effective server density remains 𝜇. We also choose 𝑐 = 1 . ASingle Resource Unit-service (SRU) system is a single resource system with server density 𝜇 and 𝑐 = 1 . A SingleResource Bulk-service (SRB) system is also a single resource system with server density 𝜇 ∕2 and 𝑐 = 2 . Note thatthe request density and effective server densities ( 𝑐𝜇 ) are same in all settings. From Figure 6(f), it is clear that TRHperforms better than TRN across all server distributions. This advocates for maintaining similar densities for eachresource type in a two resource system. As expected, a deterministic equidistant placement of servers produce theleast expected request distance for each system among all other choice of placements. SRB in deterministic serverplacement scenario performs the best among all other settings. However, it does not perform well with other serverdistributions. Also, note that, TRH has a higher expected request distance as compared to SRU across all serverdistributions. Thus Equation (61) in Section 7.5 holds true even under non-markovian setting. We consider the case in which both users and servers are distributed according to Poisson processes. From Figure 7(a), we observe that due to its directional nature MTR has a larger expected request distance compared to other policieswhile GS provides near optimal performance. At low loads i.e. when 𝜌 ≪ , the Nearest Neighbor policy policyperforms similar to the optimal policy. But as 𝜌 → , the NN policy perform worse.In Figure 7 (b), we compare the performance of allocation policies across different server capacities. The expectedrequest distance decreases with increase in server capacities across all policies. NN, GS and the optimal policy converge Panigrahy et al.:
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Page 20 of 28ne-dimensional Distributed Service Networks E xp ec t e d r e qu e s t d i s t a n ce MTRNNGSOPT (a)
Server Capacity E xp ec t e d R e qu e s t D i s t a n ce MTRNNGSOPT (b) E xp ec t e d c o s t MTRNNGSOPT (c)
Server Capacity E xp ec t e d c o s t MTRNNGSOPT (d)
Figure 7:
Comparison of different allocation policies: (a) 𝜌𝜌𝜌 vs 𝔼 [ 𝐷 ] 𝔼 [ 𝐷 ] 𝔼 [ 𝐷 ] with 𝑐 = 1 𝑐 = 1 𝑐 = 1 , (b) 𝑐𝑐𝑐 vs. 𝔼 [ 𝐷 ] 𝔼 [ 𝐷 ] 𝔼 [ 𝐷 ] with 𝜌 = 0 . 𝜌 = 0 . 𝜌 = 0 . , (c) 𝜌𝜌𝜌 vs 𝑇𝑇𝑇 with 𝛽 = 2 , 𝑡 = 1 , 𝑐 = 1 𝛽 = 2 , 𝑡 = 1 , 𝑐 = 1 𝛽 = 2 , 𝑡 = 1 , 𝑐 = 1 and (d) 𝑐𝑐𝑐 vs. 𝑇𝑇𝑇 with 𝛽 = 2 , 𝑡 = 1 , 𝜌 = 0 . 𝛽 = 2 , 𝑡 = 1 , 𝜌 = 0 . 𝛽 = 2 , 𝑡 = 1 , 𝜌 = 0 . . to the same value as 𝑐 gets higher.We now consider the expected communication cost as the performance metric. We use a cost model described inSection 7.4 with the parameter 𝛽 = 2 and 𝑡 = 1 . From Figure 7 (c), we observe that while at low loads i.e. when 𝜌 ≪ , GS and NN perform similar to the optimal policy, as 𝜌 increases both GS and NN perform worse. Note that,NN has a higher expected request distance than GS at high load as shown in Figure 7 (a). However, the performance isreversed with 𝛽 = 2 , i.e. NN has a lower expected cost than GS at high load as shown in Figure 7 (c). This depicts theeffect of 𝛽 on the performance of various allocation policies. In Figure 7 (d), we observe that NN, GS and the optimalpolicy converge to the same value as 𝑐 gets higher.We observe similar trends in the case of deterministic inter-server distance distributions. However, under equaldensities, all the policies produce smaller expected request distance as compared to their Poisson counterpart. Thisadvocates for placing equidistant servers in a bidirectional system with Poisson distributed requesters to minimizeexpected request distance.
10. Allocating Resources in 2D
We now consider the case where requesters and servers are located on the two-dimensional plane, ℝ . The problemof minimizing the expected request distance can be solved by first constructing a complete 𝑅 × 𝑆 bipartite graph withedge weights 𝑤 𝑟,𝑠 = ‖ 𝑟, 𝑠 ‖ , 𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆 ; followed by executing the Hungarian matching algorithm whose timecomplexity is 𝑂 ( 𝑛 ) , where 𝑛 = | 𝑅 | + | 𝑆 | . In this section, we present a heuristic algorithm, which leverages theoptimal dynamic programming scheme for one-dimensional inputs to solve the two-dimensional problem, has 𝑂 ( 𝑛 ) time complexity, and empirically yields request distances within a constant factor of the optimal solution.The key insight is to embed the points denoting 𝑅, 𝑆 ⊂ ℝ into new locations in ℝ such that the distances betweena requester 𝑟 ∈ 𝑅 and its 𝐾 nearest neighbors (servers) 𝑠 ∈ 𝑁𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 ( 𝑟 ) are approximately preserved. We observethat while distance-preserving or even low-distortion embeddings into a very low dimensional space like ℝ typicallydo not exist, embeddings that preserve distances to 𝐾 nearest neighbors of the other node type (for not too large 𝐾 )may be plausible. This is useful because preserving the nearest servers from ℝ to ℝ provides a reasonable opportunityfor the Dynamic Programming algorithm outlined in Section 8 to find good matchings.We achieve the aforementioned embedding by adapting a non-linear dimensionality reduction method such asLocally Linear Embedding (LLE) [19], which consists of the steps outlined below. Estimation of nearest neighbor weights.
For each requester 𝑟 𝑖 , select 𝐾 nearest servers 𝑠 𝑖 , 𝑠 𝑖 , … , 𝑠 𝑖𝐾 . Estimatea set of weights 𝑤 𝑖 , 𝑤 𝑖 , … , 𝑤 𝑖𝐾 such that the point 𝑟 𝑖 can be reconstructed from 𝑠 𝑖 , 𝑠 𝑖 , … , 𝑠 𝑖𝐾 ∶ 𝑟 𝑖 = ∑ 𝐾𝑗 =1 𝑤 𝑖𝑗 𝑠 𝑖𝑗 .Similarly for each server 𝑠 𝑖 , select 𝐾 ′ nearest requesters 𝑟 𝑖 , 𝑟 𝑖 , … , 𝑟 𝑖𝐾 ′ and estimated weights such that point 𝑠 𝑖 canbe reconstructed from the nearest neighbor requester locations: 𝑠 𝑖 = ∑ 𝐾𝑗 =1 𝑤 𝑖𝑗 𝑟 𝑖𝑗 .This can be achieved by minimizing the reconstruction error for each node 𝑖 ∈ 𝑅 × 𝑆 . Suppose 𝐾 is fixed for bothrequesters and servers. 𝑊 is an 𝑛 × 𝑛 matrix of weights where 𝑛 = | 𝑅 | + | 𝑆 | and the 𝑖 -th row 𝑊 𝑖 , which corresponds While the specific case where the weights are Euclidean distances can be solved by Agrawal’s algorithm in 𝑂 ( 𝑛 𝜖 ) time, for a more generalweight function the more expensive Hugarian algorithm is needed. Note that in general, the nearest servers need not be the ones with the smallest Euclidean distance from 𝑟 𝑖 ; they could be the ones with low costs to 𝑟 𝑖 . However in this section, we equate the costs with the Euclidean distance. Panigrahy et al.:
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Figure 8:
Approximate nearest-K-distance preserving ℝ → ℝ embedding ( 𝐾 = 25% ) to node 𝑖 , has 𝐾 non-zero elements. The structure of 𝑊 is as follows: 𝑊 = ( 𝑊 𝑅𝑆 𝑊 𝑆𝑅 ) , where 𝑊 𝑅𝑆 and 𝑊 𝑆𝑅 are rectangular matrices with dimensions | 𝑅 | × | 𝑆 | and | 𝑆 | × | 𝑅 | , respectively. If the two-dimensional coordinates of node 𝑖 are represented by vector 𝐱 𝐢 , the reconstruction error can be defined by: 𝜖 ( 𝑊 𝑖 ) = ‖ 𝐱 𝐢 − 𝐾 ∑ 𝑗 =1 𝑊 𝑖𝑗 𝐱 𝐣 ‖ . (64)It was shown in [19] that 𝜖 ( 𝑊 𝑖 ) is minimized when 𝑊 𝑖 = ( 𝐺 𝑖 + 𝜆𝐼 ) −1 , where 𝐺 𝑖 ( 𝑗, 𝑘 ) = ( 𝐱 𝐣 − 𝐱 𝐢 ) . ( 𝐱 𝐤 − 𝐱 𝐢 ) , isthe vector of all ones, and 𝜆 is chosen such that the elements of 𝑊 𝑖 add up to 1. Computing optimal embedding in ℝ . LLE suggests that the relationships between the 𝑛 points in the higher dimen-sional space ( 𝐑 in our case) captured by the matrix 𝑊 should be approximately preserved in the lower dimensionalspace ( 𝐑 in our case). Then the optimal embedding 𝐲 = { 𝑦 , 𝑦 , … , 𝑦 𝑛 } , 𝑦 𝑖 ∈ ℝ can be found by solving the followingquadratic optimization problem: min ∑ 𝑛𝑖 =1 ( 𝑦 𝑖 − ∑ 𝑗 𝑊 𝑖𝑗 𝑦 𝑗 ) = 𝐲 𝑇 ( 𝐼 − 𝑊 ) 𝑇 ( 𝐼 − 𝑊 ) 𝐲 , subject to: 𝐲 𝑇 𝐲 = 1 (65) ( 𝐼 − 𝑊 ) 𝑇 ( 𝐼 − 𝑊 ) is a positive semi-definite sparse matrix (since 𝐾 << 𝑛 ) and the optimal solution to this “eigenvalue"problem is given by the eigenvector corresponding to the smallest non-zero eigenvalue of ( 𝐼 − 𝑊 ) 𝑇 ( 𝐼 − 𝑊 ) [19]. Sincewe do not need to compute all the eigenvectors, the second smallest eigenvalue of a matrix can be computed efficientlywithout performing a matrix diagonalization using the Arnoldi algorithm in running time 𝑂 ( 𝑛 ) . Using the ℝ -embedding for matching. After generating the embedding 𝐲 , we applied our Dynamic ProgrammingAlgorithm to compute the best resource allocation scheme. However, naive application of the algorithm led to high ex-pected request distances. The reason behind this is illustrated in Figure 8, which visualizes an embedding computed fora given set of requesters and servers from ℝ to ℝ ; the zigzag lines denote the linear order imposed by the embedding. Panigrahy et al.:
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Figure 9:
Matching for a clustered distribution: | 𝑅 | = 50 requesters are spread uniformly at random in a square [0 , , and each of the | 𝑆 | = 100 servers is located uniformly at random in a box of size . around a randomly selectedresource. (a) Optimum weighted bipartite matching in 2D ( 𝔼 [ 𝐷 ] = 0 . ); (b) Approximate matching after 1D embedding( 𝔼 [ 𝐷 ] = 0 . ) ( 𝐾 = 25%) | 𝑅 | | 𝑆 | 𝔼 [ 𝐷 ] 𝔼 [ 𝐷 ] Optimum ℝ → ℝ
200 400 0 .
023 0 . . . .
018 0 . . . . . . . .
014 0 . Table 2
Matching in larger networks: requesters are spread uniformly at random in a square [0 ,
1] ×[0 , and each server is located uniformly at random in a box of size . around a randomlyselected resource. It is easy to see that it is quite possible that a pair of (requester (blue), server (orange)) nodes which are far away in ℝ may be pretty close to each other in ℝ . Since our LLE-based scheme only tries to preserve close-by neighbors anddoes not explicitly attempt to repel nodes that are farther away in ℝ , such pairs of nodes could end up being embeddedclose to each other in ℝ . To circumvent this problem, we propose a heuristic scheme to adjust the embedding 𝐲 suchthat whenever for a pair of nodes { 𝑖, 𝑗 } we have ‖ 𝐱 𝐢 − 𝐱 𝐣 ‖ > Δ but ‖ 𝑦 𝑘 − 𝑦 𝑘 +1 ‖ < 𝜖 , where 𝐱 𝐢 is mapped to 𝑦 𝑘 and 𝐱 𝐣 is mapped to 𝑦 𝑘 +1 , and Δ , 𝜖, 𝛿 are configurable constants, we increase the distance between 𝑦 𝑘 and 𝑦 𝑘 +1 by adding alarge cumulative constant 𝑐 𝑘 +1 = 𝑐 𝑘 + 𝛿 to 𝑦 𝑘 +1 . This adjustment of 𝐲 sequentially spreads out the points in ℝ towardthe right and the Dynamic Programing Algorithm is then able to find good requester-server matchings.Figure 9 shows a comparison between an optimum matching and an approximate matching constructed by theembedding methods proposed in this section. Table 2 shows results for the case when | 𝑆 | is varied for a fixed | 𝑅 | . Wecan observe that the ℝ → ℝ embedding approach yields a solution which is empirically close to 𝑂𝑃 𝑇 . Giventhat this procedure has a lower time complexity 𝑂 ( 𝑛 ) than the usual 𝑂 ( 𝑛 ) for Hungarian algorithm, it could bepractically useful for large resource allocation problems. Both the embedding process and the Dynamic Programming algorithm have time complexity 𝑂 ( 𝑛 ) . Panigrahy et al.:
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11. Conclusion
We introduced a queuing theoretic model for analyzing the behavior of unidirectional policies to allocate tasksto servers on the real line. We showed the equivalence of UGS and MTR w.r.t the expected request distance andpresented results associated with the case when either requesters or servers were Poisson distributed. In this context, weanalyzed a new queueing theoretic model: ESABQ, not previously studied in queueing literature. We also proposed adynamic programming based algorithm to obtain an optimal allocation policy in a bi-directional system. We performedsensitivity analysis for unidirectional system and compared the performance of various greedy allocation strategiesalong with the unidirectional policies to that of optimal policy. We proposed a heuristic based approximate solutionto the optimal assignment problem for the two-dimensional scenario. Going further, we aim to extend our analysis forunidirectional policies to a two-dimensional geographic region.
12. Acknowledgment
This research was sponsored by the U.S. Army Research Laboratory and the U.K. Defence Science and TechnologyLaboratory under Agreement Number W911NF-16-3-0001 and by the NSF under grant NSF CNS-1617437. The viewsand conclusions contained in this document are those of the authors and should not be interpreted as representing theofficial policies, either expressed or implied, of the U.S. Army Research Laboratory, the U.S. Government, the U.K.Defence Science and Technology Laboratory. This document does not contain technology or technical data controlledunder either the U.S. International Traffic in Arms Regulations or the U.S. Export Administration Regulations.
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13. Appendix 𝐹 𝑍 for various inter-server distance distrbutions 𝐹 𝑋 ( 𝑥 ) ∼ Exponential ( 𝜇 ) 𝐹 𝑋 ( 𝑥 ) ∼ Exponential ( 𝜇 ) 𝐹 𝑋 ( 𝑥 ) ∼ Exponential ( 𝜇 ) In this case, both 𝑋 and 𝑌 are exponentially distributed. Thus the difference distribution is given by 𝐷 𝑋𝑌 ( 𝑥 ) = 1 − 𝜆𝜆 + 𝜇 𝑒 − 𝜇𝑥 , when 𝑥 ≥ (66)Combining (9) and (66), we get 𝐹 𝑍 ( 𝑥 ) = 1 − 𝜆𝜆 + 𝜇 𝑒 − 𝜇𝑥 − 1 + 𝜆𝜆 + 𝜇𝜆𝜆 + 𝜇 = 1 − 𝑒 − 𝜇𝑥 . (67)Thus we obtain 𝐹 𝑋 ( 𝑥 ) = 𝐹 𝑍 ( 𝑥 ) ∼ Exponential ( 𝜇 ) . 𝐹 𝑋 ( 𝑥 ) ∼ Uniform (0 , 𝑏 ) 𝐹 𝑋 ( 𝑥 ) ∼ Uniform (0 , 𝑏 ) 𝐹 𝑋 ( 𝑥 ) ∼ Uniform (0 , 𝑏 ) The c.d.f. for uniform distribution is 𝐹 𝑋 ( 𝑥 ) = { 𝑥𝑏 , ≤ 𝑥 ≤ 𝑏 ;1 , 𝑥 > 𝑏, (68)where 𝑏 is the uniform parameter. Thus we have 𝐷 𝑋𝑌 ( 𝑥 ) = ∫ ∞0 𝐹 𝑋 ( 𝑥 + 𝑦 ) 𝜆𝑒 − 𝜆𝑦 𝑑𝑦 = [ ∫ 𝑏 − 𝑥 𝑥 + 𝑦𝑏 𝜆𝑒 − 𝜆𝑦 𝑑𝑦 ] + [ ∫ ∞ 𝑏 − 𝑥 𝜆𝑒 − 𝜆𝑦 𝑑𝑦 ] = 𝜆𝑥 − 𝑒 − 𝜆 ( 𝑏 − 𝑥 ) + 𝑒 − 𝜆𝑏 𝑏𝜆 + 𝑒 − 𝜆𝑏 − 1 (69)Taking 𝑘 𝜆 = 1∕( 𝑏𝜆 + 𝑒 − 𝜆𝑏 − 1) and using Equation (9) we have 𝐹 𝑍 ( 𝑥 ) = 𝑘 𝜆 [ 𝜆𝑥 + 𝑒 − 𝜆𝑏 (1 − 𝑒 𝜆𝑥 ) ] and 𝑓 𝑍 ( 𝑥 ) = 𝜆𝑘 𝜆 [ 𝑒 − 𝜆𝑏 𝑒 𝜆𝑥 ) ] . (70)Taking 𝛼 𝑍 = ∫ 𝑏 𝑥𝑓 𝑍 ( 𝑥 ) 𝑑𝑥 and 𝜎 𝑍 = [ ∫ 𝑏 𝑥 𝑓 𝑍 ( 𝑥 ) 𝑑𝑥 ] − 𝛼 𝑍 we have 𝛼 𝑍 = 𝑏 𝜆 𝑘 𝜆 − 1 𝜆 , 𝜎 𝑍 = 𝑏 𝜆 𝑘 𝜆 − 𝑘 𝜆 𝜆 [ 𝑏 ( 𝑏𝜆 − 2) + 2 𝜆 (1 − 𝑒 − 𝜆𝑏 ) ] − 𝛼 𝑍 , 𝛼 𝑋 = 𝑏 ∕2 , 𝜎 𝑋 = 𝑏 ∕12 . (71) 𝐹 𝑋 ( 𝑥 ) ∼ Deterministic ( 𝑑 ) 𝐹 𝑋 ( 𝑥 ) ∼ Deterministic ( 𝑑 ) 𝐹 𝑋 ( 𝑥 ) ∼ Deterministic ( 𝑑 ) Another interesting scenario is when servers are equally spaced at a distance 𝑑 from each other i.e. when 𝐹 𝑋 ( 𝑥 ) ∼ Deterministic ( 𝑑 ) . The c.d.f. for deterministic distribution is 𝐹 𝑋 ( 𝑥 ) = { , ≤ 𝑥 < 𝑑 ;1 , 𝑥 ≥ 𝑑 , (72)where 𝑑 is the deterministic parameter. A similar analysis as that of uniform distribution yields 𝐹 𝑍 ( 𝑥 ) = 𝑐 𝜆 [ 𝑒 − 𝜆 ( 𝑑 − 𝑥 ) − 𝑒 𝜆𝑑 ] ; 𝑓 𝑍 ( 𝑥 ) = 𝜆𝑐 𝜆 [ 𝑒 − 𝜆 ( 𝑑 − 𝑥 ) ] , (73) where 𝑐 𝜆 = 1∕(1 − 𝑒 − 𝜆𝑑 ) . Thus we have 𝛼 𝑍 = 𝑐 𝜆 𝑑 𝜆 + 𝑒 − 𝜆𝑑 − 1 𝜆 , 𝜎 𝑍 = 𝑐 𝜆 𝜆 [ 𝑑 ( 𝑑 𝜆 − 2) + 2 𝜆 (1 − 𝑒 − 𝜆𝑑 ) ] − 𝛼 𝑍 , 𝛼 𝑋 = 𝑑 , 𝜎 𝑋 = 0 . (74) Panigrahy et al.:
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Let us write the Chapman-Kolmorogov equations for the Markov chain {( 𝐿 ( 𝑡 ) , 𝑅 ( 𝑡 ) , 𝐼 ( 𝑡 )) , 𝑡 ≥ defined in Section6.3.1.For 𝑛 ≥ and 𝑥 > we get 𝜕𝜕𝑡 𝑝 𝑡 ( 𝑛, 𝑥 ; 1) = 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑛, 𝑥 ; 1) − 𝜆𝑝 𝑡 ( 𝑛, 𝑥 ; 1) − 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑛,
0; 1) + 𝜆𝑝 𝑡 ( 𝑛 − 1 , 𝑥 ; 1) 𝜕𝜕𝑡 𝑝 𝑡 ( 𝑛, 𝑥 ; 2) = 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑛, 𝑥 ; 2) − 𝜆𝑝 𝑡 ( 𝑛, 𝑥 ; 2) − 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑛,
0; 2) + 𝜆𝑝 𝑡 ( 𝑛 − 1 , 𝑥 ; 2) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑛 + 𝑐,
0; 1) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑝 𝑡 ( 𝑛 + 𝑐,
0; 2) . Letting 𝑡 → ∞ yields 𝜕𝜕𝑥 𝑝 ( 𝑛, 𝑥 ; 1) − 𝜆𝑝 ( 𝑛, 𝑥 ; 1) − 𝜕𝜕𝑥 𝑝 ( 𝑛,
0; 1) + 𝜆𝑝 ( 𝑛 − 1 , 𝑥 ; 1) (75) 𝜕𝜕𝑥 𝑝 ( 𝑛, 𝑥 ; 2) − 𝜆𝑝 ( 𝑛, 𝑥 ; 2) − 𝜕𝜕𝑥 𝑝 ( 𝑛,
0; 2) + 𝜆𝑝 ( 𝑛 − 1 , 𝑥 ; 2) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑝 ( 𝑛 + 𝑐,
0; 1) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑝 ( 𝑛 + 𝑐,
0; 2) . (76)For 𝑛 = 1 , 𝑥 > 𝜕𝜕𝑡 𝑝 𝑡 (1 , 𝑥 ; 1) = 𝜕𝜕𝑥 𝑝 𝑡 (1 , 𝑥 ; 1) − 𝜆𝑝 𝑡 (1 , 𝑥 ; 1) − 𝜕𝜕𝑥 𝑝 𝑡 (1 ,
0; 1) + 𝜆𝑝 𝑡 (0) 𝐹 𝑍 ( 𝑥 ) 𝜕𝜕𝑡 𝑝 𝑡 (1 , 𝑥 ; 2) = 𝜕𝜕𝑥 𝑝 𝑡 (1 , 𝑥 ; 2) − 𝜆𝑝 𝑡 (1 , 𝑥 ; 2) − 𝜕𝜕𝑥 𝑝 𝑡 (1 ,
0; 2) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑝 (1 + 𝑐,
0; 1) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑝 𝑡 (1 + 𝑐,
0; 2) . Letting 𝑡 → ∞ yields 𝜕𝜕𝑥 𝑝 (1 , 𝑥 ; 1) − 𝜆𝑝 (1 , 𝑥 ; 1) − 𝜕𝜕𝑥 𝑝 (1 ,
0; 1) + 𝜆𝑝 (0) 𝐹 𝑍 ( 𝑥 ) (77) 𝜕𝜕𝑥 𝑝 (1 , 𝑥 ; 2) − 𝜆𝑝 (1 , 𝑥 ; 2) − 𝜕𝜕𝑥 𝑝 (1 ,
0; 2) + 𝐹 𝑋 ( 𝑥 ) ( 𝜕𝜕𝑥 𝑝 (1 + 𝑐,
0; 1) + 𝜕𝜕𝑥 𝑝 (1 + 𝑐,
0; 2) ) , 𝑥 > . (78)We can collect the results in (75)-(78) as follows: for 𝑛 ≥ , 𝑥 > , 𝜕𝜕𝑥 𝑝 ( 𝑛, 𝑥 ; 1) − 𝜆𝑝 ( 𝑛, 𝑥 ; 1) − 𝜕𝜕𝑥 𝑝 ( 𝑛,
0; 1) + 𝜆𝑝 ( 𝑛 − 1 , 𝑥 ; 1) ( 𝑛 ≥
2) + 𝜆𝑝 (0) 𝐹 𝑍 ( 𝑥 ) ( 𝑛 = 1) (79) 𝜕𝜕𝑥 𝑝 ( 𝑛, 𝑥 ; 2) − 𝜆𝑝 ( 𝑛, 𝑥 ; 2) − 𝜕𝜕𝑥 𝑝 ( 𝑛,
0; 2) + 𝜆𝑝 ( 𝑛 − 1 , 𝑥 ; 2) ( 𝑛 ≥
2) + 𝐹 𝑋 ( 𝑥 ) ( 𝜕𝜕𝑥 𝑝 ( 𝑛 + 𝑐,
0; 1) + 𝜕𝜕𝑥 𝑝 ( 𝑛 + 𝑐,
0; 2) ) . (80)Define 𝑔 ( 𝑛, 𝑥 ) = 𝑝 ( 𝑛, 𝑥 ; 1) + 𝑝 ( 𝑛, 𝑥 ; 2) for 𝑛 ≥ , 𝑥 > . Summing (79) and (80) gives 𝜕𝜕𝑥 𝑔 ( 𝑛, 𝑥 ) − 𝜆𝑔 ( 𝑛, 𝑥 ) − 𝜕𝜕𝑥 𝑔 ( 𝑛,
0) + 𝜆𝑔 ( 𝑛 − 1 , 𝑥 ) ( 𝑛 ≥
2) + 𝜆𝑝 (0) 𝐹 𝑍 ( 𝑥 ) ( 𝑛 = 1) + 𝐹 𝑋 ( 𝑥 ) 𝜕𝜕𝑥 𝑔 ( 𝑛 + 𝑐, , ∀ 𝑛 ≥ , 𝑥 > . (81) 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) Assume that 𝐹 𝑋 ( 𝑥 ) = 1 − 𝑒 − 𝜇𝑥 (regular batch service times are exponentially distributed). Then, 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) = − 𝜌𝑧 𝑐 +1 + (1 + 𝜌 ) 𝑧 𝑐 − 11 + 𝜌 (1 − 𝑧 ) .𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) = 0 for | 𝑧 | ≤ iff 𝑄 ( 𝑧 ) ∶= − 𝜌𝑧 𝑐 +1 + (1 + 𝜌 ) 𝑧 𝑐 − 1 = 0 . The derivative of 𝑄 ( 𝑧 ) is 𝑄 ′ ( 𝑧 ) = 𝑧 𝑐 −1 ((1 + 𝜌 ) 𝑐 − 𝜌 ( 𝑐 + 1) 𝑧 ) . It vanishes at 𝑧 = 0 and at 𝑧 = (1+ 𝜌 ) 𝑐𝜌 ( 𝑐 +1) > under the stability condition 𝜌 < 𝑐 . Since 𝑧 = 0 is not a zero of 𝑄 ( 𝑧 ) , we conclude that all zeros of 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) in { | 𝑧 | ≤ have multiplicity one.More generally, it is shown in [6] that all zeros of 𝑧 𝑐 − 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) in { | 𝑧 | ≤ have multiplicity one if 𝐹 𝑋 is a 𝜒 -distribution with an even number 𝑝 of degrees of freedom, i.e. 𝑑𝐹 𝑋 ( 𝑥 ) = 𝑎 𝑝 Γ( 𝑝 ) 𝑥 𝑝 −1 𝑒 − 𝑎𝑥 𝑑𝑥 so that 𝜇 = 𝑝 ∕ 𝑎 . Panigrahy et al.:
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Uncrossing an assignment either reduces request distance or keeps it unchanged. 𝐴 ( 𝑧 ) Define 𝐴 ( 𝑧 ) = 𝐹 ∗ 𝑋 ( 𝜃 ( 𝑧 )) . If 𝐴 ( 𝑧 ) has a radius of convergence larger than one (i.e. 𝐴 ( 𝑧 ) is analytic for | 𝑧 | ≤ 𝜈 with 𝜈 > ) and 𝐴 ′ (1) < 𝑐 ∈ {1 , , …} a direct application of Rouché’s theorem shows that 𝑧 𝑐 − 𝐴 ( 𝑧 ) has 𝑐 zeros in theunit disk { | 𝑧 | ≤ (see e.g. [2]). If the radius of convergence of 𝐴 ( 𝑧 ) is one, 𝐴 ( 𝑧 ) is differentiable at 𝑧 = 1 , 𝐴 ′ (1) < 𝑐 ,and 𝑧 𝑐 − 𝐴 ( 𝑧 ) has period 𝑝 , then 𝑧 𝑐 − 𝐴 ( 𝑧 ) has exactly 𝑝 ≤ 𝑠 zeros on the unit circle and 𝑠 − 𝑝 zeros inside the unitdisk { | 𝑧 | < [2, Theorem 3.2]. Assume that the stability condition 𝑑𝑑𝑧 𝐴 ( 𝑧 ) | 𝑧 =1 = 𝜌 < 𝑐 holds. 𝐴 ( 𝑧 ) has a radius ofconvergence larger than one when 𝐹 𝑋 is the exponential/Erlang/Gamma/ etc probability distributions. One easily checks that (31) gives the classical Pollaczek-Khinchin formula for the M/G/1 queue when 𝑐 = 1 and 𝐹 𝑍 = 𝐹 𝑋 .Let now 𝑐 = 1 in (31) with 𝐹 𝑍 and 𝐹 𝑋 arbitrary. Then, 𝑁 ( 𝑧 ) = 𝑎 𝜆 ( 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) − 𝑧𝐹 ∗ 𝑍 ( 𝜆 (1 − 𝑧 )) 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) − 𝑧 ) gives the 𝑧 -transform of the stationary number of customers in a M/G/1 queue with an exceptional first customer ina busy period. The constant 𝑎 ∕ 𝜆 is obtained from the identity 𝑁 (1) = 1 by application of L’Hopital’s rule, whichgives 𝑎 ∕ 𝜆 = (1 − 𝜌 )∕(1 − 𝜌 + 𝜌 𝑍 ) . This gives 𝑁 ( 𝑧 ) = 1 − 𝜌 𝜌 + 𝜌 𝑍 ( 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) − 𝑧𝐹 ∗ 𝑍 ( 𝜆 (1 − 𝑧 )) 𝐹 ∗ 𝑋 ( 𝜆 (1 − 𝑧 )) − 𝑧 ) . The above is a known result [22].If 𝐹 ∗ 𝑍 = 𝐹 ∗ 𝑋 ∶= 𝐹 ∗ , then 𝑁 ( 𝑧 ) = ∑ 𝑐𝑘 =1 𝑎 𝑘 [ ( 𝑧 𝑐 − 𝑧 𝑘 ) 𝑧 𝑐 + ((1 − 𝑧 𝑐 ) 𝑧 − (1 − 𝑧 𝑘 )) 𝐹 ∗ ( 𝜃 ( 𝑧 )) ] 𝜃 ( 𝑧 )( 𝑧 𝑐 − 𝐹 ∗ ( 𝜃 ( 𝑧 )) . 𝑣 ( 𝑧 ) and 𝑣 ( 𝑧 ) 𝑣 ( 𝑧 ) in (43) can further be simplified to 𝑣 ( 𝑧 ) = ∞ ∑ 𝑙 =0 𝑧 𝑙 𝑙 ∑ 𝑚 =0 𝜋 𝑚 𝑐 ∑ 𝑖 =0 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 = ∞ ∑ 𝑚 =0 𝜋 𝑚 ∑ 𝑙 ≥ 𝑚 𝑧 𝑙 𝑐 ∑ 𝑖 =0 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 = ∞ ∑ 𝑚 =0 𝜋 𝑚 𝑧 𝑚 ∑ 𝑙 ≥ 𝑚 𝑧 𝑙 − 𝑚 𝑐 ∑ 𝑖 =0 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 = ∞ ∑ 𝑚 =0 𝜋 𝑚 𝑧 𝑚 ∞ ∑ 𝑗 =0 𝑧 𝑗 𝑐 ∑ 𝑖 =0 𝑘 𝑖 + 𝑗 𝑝 𝑖 Note that we retrieve this result by letting 𝑐 = 1 in (32). Panigrahy et al.:
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Page 27 of 28ne-dimensional Distributed Service Networks = 𝑁 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑝 𝑖 𝑧 − 𝑖 ∞ ∑ 𝑗 =0 𝑧 𝑖 + 𝑗 𝑘 𝑖 + 𝑗 = 𝑁 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑝 𝑖 𝑧 − 𝑖 [ 𝐾 ( 𝑧 ) − 𝑖 ∑ 𝑗 =0 𝑘 𝑗 𝑧 𝑗 + 𝑘 𝑖 𝑧 𝑖 ] = 𝑁 ( 𝑧 ) { 𝑐 ∑ 𝑖 =0 𝑝 𝑖 𝑧 − 𝑖 [ 𝐾 ( 𝑧 ) − 𝑖 ∑ 𝑗 =0 𝑘 𝑗 𝑧 𝑗 ] + 𝑐 ∑ 𝑖 =0 𝑘 𝑖 𝑧 𝑖 } . (82) Similarly 𝑣 ( 𝑧 ) in (44) can further be simplified to 𝑣 ( 𝑧 ) = ∞ ∑ 𝑙 =0 𝑧 𝑙 𝑐 + 𝑙 ∑ 𝑚 = 𝑙 +1 𝜋 𝑚 𝑐 ∑ 𝑖 = 𝑚 − 𝑙 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 = [ ∞ ∑ 𝑙 =0 𝑧 𝑙 𝑐 + 𝑙 ∑ 𝑚 = 𝑙 𝜋 𝑚 𝑐 ∑ 𝑖 = 𝑚 − 𝑙 𝑘 𝑖 + 𝑙 − 𝑚 𝑝 𝑖 ] − 𝑁 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑘 𝑖 𝑧 𝑖 = [ 𝑐 ∑ 𝑚 =0 𝑧 − 𝑚 𝑐 ∑ 𝑖 = 𝑚 𝑘 𝑖 − 𝑚 𝑝 𝑖 ∞ ∑ 𝑙 =0 𝑧 𝑚 + 𝑙 𝜋 𝑚 + 𝑙 ] − 𝑁 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑘 𝑖 𝑧 𝑖 = [ 𝑐 ∑ 𝑚 =0 𝑧 − 𝑚 𝑐 ∑ 𝑖 = 𝑚 𝑘 𝑖 − 𝑚 𝑝 𝑖 { 𝑁 ( 𝑧 ) − 𝑚 −1 ∑ 𝑗 =0 𝜋 𝑗 𝑧 𝑗 }] − 𝑁 ( 𝑧 ) 𝑐 ∑ 𝑖 =0 𝑘 𝑖 𝑧 𝑖 . (83) 𝐴 ( 𝑧 ) Denote 𝐴 ( 𝑧 ) = 𝐾 ( 𝑧 ) ∑ 𝑐𝑖 =0 𝑝 𝑐 − 𝑖 𝑧 𝑖 . Clearly, 𝐴 ( 𝑧 ) is also a probability generating function ( pgf ) for the non-negativerandom variable 𝑉 + ̃ where ̃ is a random variable on {0 , … , 𝑐 − 1} with distribution Pr ( ̃ = 𝑗 ) = 𝑝 𝑐 − 𝑗 , ∀ 𝑗 ∈{0 , , … , 𝑐 − 1} . Also we have 𝐴 ′ (1) = 𝐾 ′ (1) + 𝑐 ∑ 𝑖 =0 𝑝 𝑐 − 𝑖 𝑖 = 𝜌 + 𝑐 ∑ 𝑖 =1 𝑝 𝑖 ( 𝑐 − 𝑖 ) = 𝜌 + 𝑐 ∑ 𝑖 =1 𝑝 𝑖 𝑐 − 𝑐 ∑ 𝑖 =1 𝑖𝑝 𝑖 = 𝜌 + 𝑐 − From our stability condition we know that 𝜌 < . Thus 𝐴 ′ (1) < 𝑐. Since 𝐴 ( 𝑧 ) is a pgf and 𝐴 ′ (1) < 𝑐 , by applying thearguments from [2, Theorem 3.2] we conclude that the denominator of equation (49) has 𝑐 − 1 zeros inside and one onthe unit circle, | 𝑧 | = 1 . Proof.
It can be observed that if such a 4-tuple ( 𝑖, 𝑗, 𝑖 ′ , 𝑗 ′ ) exists, the cost can be reduced by assigning 𝑖 to 𝑗 ′ and 𝑖 ′ to 𝑗 , hence we arrive at a contradiction. To show this, consider the six possible cases of relative ordering between 𝑟 𝑖 , 𝑟 𝑖 ′ , 𝑠 𝑗 , 𝑠 𝑗 ′ which obey 𝑟 𝑖 < 𝑟 𝑖 ′ and 𝑠 𝑗 > 𝑠 𝑗 ′ . We give a pictorial proof in Figure 10 . It is easy to see that in each ofthe cases, the request distance of the uncrossed assignment is either smaller or remains unchanged. For ease of exposition, the requesters and servers are shown to be located along two separate horizontal lines, although they are located on thesame real-line.
Panigrahy et al.: