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Research article Special Issues

Analysis of single-cell RNA-sequencing data identifies a hypoxic tumor subpopulation associated with poor prognosis in triple-negative breast cancer


  • Triple-negative breast cancer (TNBC) is an aggressive subtype of mammary carcinoma characterized by low expression levels of estrogen receptor (ER), progesterone receptor (PR), and human epidermal growth factor receptor 2 (HER2). Along with the rapid development of the single-cell RNA-sequencing (scRNA-seq) technology, the heterogeneity within the tumor microenvironment (TME) could be studied at a higher resolution level, facilitating an exploration of the mechanisms leading to poor prognosis during tumor progression. In previous studies, hypoxia was considered as an intrinsic characteristic of TME in solid tumors, which would activate downstream signaling pathways associated with angiogenesis and metastasis. Moreover, hypoxia-related genes (HRGs) based risk score models demonstrated nice performance in predicting the prognosis of TNBC patients. However, it is essential to further investigate the heterogeneity within hypoxic TME, such as intercellular communications. In the present study, utilizing single-sample Gene Set Enrichment Analysis (ssGSEA) and cell-cell communication analysis on the scRNA-seq data retrieved from Gene Expression Omnibus (GEO) database with accession number GSM4476488, we identified four tumor subpopulations with diverse functions, particularly a hypoxia-related one. Furthermore, results of cell-cell communication analysis revealed the dominant role of the hypoxic tumor subpopulation in angiogenesis- and metastasis-related signaling pathways as a signal sender. Consequently, regard the TNBC cohorts acquired from The Cancer Genome Atlas (TCGA) and GEO as train set and test set respectively, we constructed a risk score model with reliable capacity for the prediction of overall survival (OS), where ARTN and L1CAM were identified as risk factors promoting angiogenesis and metastasis of tumors. The expression of ARTN and L1CAM were further analyzed through tumor immune estimation resource (TIMER) platform. In conclusion, these two marker genes of the hypoxic tumor subpopulation played vital roles in tumor development, indicating poor prognosis in TNBC patients.

    Citation: Yi Shi, Xiaoqian Huang, Zhaolan Du, Jianjun Tan. Analysis of single-cell RNA-sequencing data identifies a hypoxic tumor subpopulation associated with poor prognosis in triple-negative breast cancer[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 5793-5812. doi: 10.3934/mbe.2022271

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  • Triple-negative breast cancer (TNBC) is an aggressive subtype of mammary carcinoma characterized by low expression levels of estrogen receptor (ER), progesterone receptor (PR), and human epidermal growth factor receptor 2 (HER2). Along with the rapid development of the single-cell RNA-sequencing (scRNA-seq) technology, the heterogeneity within the tumor microenvironment (TME) could be studied at a higher resolution level, facilitating an exploration of the mechanisms leading to poor prognosis during tumor progression. In previous studies, hypoxia was considered as an intrinsic characteristic of TME in solid tumors, which would activate downstream signaling pathways associated with angiogenesis and metastasis. Moreover, hypoxia-related genes (HRGs) based risk score models demonstrated nice performance in predicting the prognosis of TNBC patients. However, it is essential to further investigate the heterogeneity within hypoxic TME, such as intercellular communications. In the present study, utilizing single-sample Gene Set Enrichment Analysis (ssGSEA) and cell-cell communication analysis on the scRNA-seq data retrieved from Gene Expression Omnibus (GEO) database with accession number GSM4476488, we identified four tumor subpopulations with diverse functions, particularly a hypoxia-related one. Furthermore, results of cell-cell communication analysis revealed the dominant role of the hypoxic tumor subpopulation in angiogenesis- and metastasis-related signaling pathways as a signal sender. Consequently, regard the TNBC cohorts acquired from The Cancer Genome Atlas (TCGA) and GEO as train set and test set respectively, we constructed a risk score model with reliable capacity for the prediction of overall survival (OS), where ARTN and L1CAM were identified as risk factors promoting angiogenesis and metastasis of tumors. The expression of ARTN and L1CAM were further analyzed through tumor immune estimation resource (TIMER) platform. In conclusion, these two marker genes of the hypoxic tumor subpopulation played vital roles in tumor development, indicating poor prognosis in TNBC patients.



    Semi-open queueing networks (SOQNs), also referred to as open queues with restricted capacity, have garnered the attention of the stochastic modeling research community in recent years; see the survey [1] and the paper [2]. SOQNs are characterized by users, who would like to receive service, arriving from an infinite population of sources and permanently departing from the network after service, as in open queueing networks. However, only a restricted number of users can receive service in the network at the same time, as in closed networks.

    A SOQN can be organized into a complex consisting of a set of orders (permissions, windows, threads, tokens, operators, robots, automated guided vehicles, etc.) required to begin service and a core (inner) network, whose nodes are dedicated to providing service for an admitted user. If an arriving user finds an available order in the set, the user enters the core network and sequentially receives service in its nodes. After the end of service in the required sequence of nodes, the user leaves the core network and returns the order, that was received at the entrance to the set of orders. If the set is empty at a user arrival instant, then the user joins a buffer and waits for an order release (this kind of service organization is called back-ordering SOQN, see [3]) or is lost (called user loss SOQN).

    As stated in [1], the utilization of SOQNs originates from the study [4] conducted by Avi-Itzhak and Heyman, who examined the efficacy of a multi-programming computer system. In that system, multiple tasks can be executed concurrently, with the central processor managing one task while peripheral devices attend to other tasks.

    Brief overviews of the research on SOQN analysis may be found, for example, in [1,5,6,7] and more recent publications [3,8,9,10,11,12,13] and references therein. Examples of possible applications of SOQNs are listed in [1], as follows: manufacturing material control rules commonly implemented in manufacturing shops to manage the inventory of both finished goods and work-in-process; modeling the operation of vehicle rental providers, where users borrow vehicles from rental depots and return them after the rental time; modeling systems for storage and retrieval based on autonomous vehicles; modeling communication networks based on window flow control; and modeling health-care systems to accurately depict the dynamics of patient-resource waiting, see also [14]. Unmanned manufacturing factories (UMFs) and robotic mobile fulfillment systems (RMFSs) are also very common applications of SOQNs; see, for example, [3,15,16,17,18,19,20,21,22,23,24].

    The vast bulk of analytical models of SOQNs currently available in the literature, as well as open and closed queueing networks, assume that the arrival flow of users is determined by the stationary Poisson process. However, it is widely acknowledged that modern real-world arrival processes are poorly fitted by such process. It may be too optimistic to predict real-world systems' performance metrics assuming that the stationary Poisson process describes the arrival flow. This is due to its constant arrival rate, relatively low variance, and zero coefficient of correlation for the sequential inter-arrival periods. Therefore, any irregularity in the arrival process, which is the inherent feature of the majority of real flows, is ignored. Namely, the existence of periods of peak rates leads to congestion and worsens the performance characteristics of the system.

    As a much more adequate model of real-world arrival processes, M. Neuts [25] introduced the versatile arrival process, later called the Markov arrival process (MAP), see [26]. More information about the MAP and its extension, such as the batch Markov arrival process (BMAP), can be found, e.g., in [27,28,29,30,31]. To our knowledge, a SOQN with the MAP is only taken into consideration in a small number of articles; see the corresponding references in [7,8,9].

    The MAP assumes that arriving users are of the same type. A more general model of the arrival process is the MMAP (marked Markov arrival process), in which arriving users are heterogeneous; see, e.g., [32]. A SOQN with multi-server nodes and the MMAP, where the type of user predefines the node in the network where the first service of the user is performed, was recently considered in [10].

    In this paper, we analyze a SOQN with the MMAP and single-server nodes. The novelty of the model, compared to all existing literature that examines SOQN, consists of the possibility of changing the rate of service in the nodes of the network depending on the number of users admitted to the network.

    The literature devoted to the analysis of various controlled queues, including control by the arrival rate, service rate, or number of active servers, is huge. This is easily explained by the fast technological development and consequent possibilities to monitor the state of a real-world system or network and dynamically change the service regime correspondingly. Such control can provide a significant economic effect due to the possibility of regime management in such a way that fast but expensive service regimes are temporarily used only in the case of high system congestion.

    Control by the service rate in the nodes can be implemented via the use of equipment with distinct performance (e.g., leasing of channels or routers with different bandwidths in telecommunication networks) or the use of various numbers of workers providing service in a node. High-performance equipment and (or) more staff can be used when the quantity of users receiving service in the network is large, which may imply a long delay of the user in the network, possible dissatisfaction with the quality of service, and possible choice of another service provider in the future. When the network is less congested, it makes sense to use lower-performance equipment and (or) less staff to spend less money on service provision.

    As early papers that examine queues with controlled service rates, we can mention [33,34]. When the system has several available service regimes listed in ascending order of service rate (and, correspondingly, increased cost), it is proved that the optimal control policy is monotone, i.e., of the multi-threshold type. This policy is defined by the set of integer numbers (thresholds). Selection of the service rate at decision moments is performed based on the relationship between the system's current user count and the predetermined thresholds. An application of such a type of policy to the BMAP/G/1 queue with a controlled service time, which can be varied at moments of service completion, was implemented in [35]. The BMAP/SM/1-type queueing system with a multi-threshold control by the service rate, semi-Markov service process, and user retrials in the case of a busy server was analyzed in [36].

    One drawback of the threshold policy is that when the current number of users in the system approaches a certain threshold, there is a chance that the service rate will fluctuate often, a phenomenon known as oscillation. To overcome this negative phenomenon (due to the possible time loss or charge for the switching), the so-called hysteresis strategy has been offered. This type of strategy assumes that each policy threshold can be split into two thresholds. The rate of service increases when the larger of these two thresholds is exceeded. When the length of a queue drops to a smaller threshold, the service slows. The system stays in the prior regime when the queue length falls between the thresholds. This delay in switching (called hysteresis in Greek) leads to a decrease in the frequency of regime switching.

    The Mx/G/1-type queueing system with two available service regimes, a batch stationary Poisson arrival process, a general distribution of service time, an account of switch-over times, and a hysteresis policy of control was considered in [37,38]. The result was generalized to the BMAP/G/1-type queueing system in [39]. An analysis of the queue with hysteresis control is more tricky than that of the same type of queue with threshold control. This is because, given the fixed values of the thresholds of the threshold policy, the choice of a service rate at a decision instant is uniquely determined by the current number of users in the network. In the case of the hysteresis policy of control, selection depends also on the service rate used before the moment of decision-making.

    In this paper, we consider a user loss SOQN with arbitrary topology, an arriving flow defined by the MMAP, the availability of a finite number L,L2, of service regimes, stochastic routing, and the hysteresis type of control by the service rates in the nodes. An exact algorithmic study of the steady-state dynamic of the network is implemented and numerically illustrated.

    The structure of the paper is as follows. The description of the SOQN under study is presented in Section 2. The topology of the network is explained, a necessary notation is presented, and the hysteresis-type control policy is defined in detail. In Section 3, a multidimensional continuous-time Markov chain defining the system behavior under the fixed set of thresholds of the control strategy is introduced. The methodology tracing back to [40] is used here for keeping track of the number of users residing in all nodes of the network. The explicit form of the infinitesimal generator of this Markov chain is derived. Formulas for the calculation of a variety of performance measures of the network via the vectors of the steady-state probabilities of the Markov chain are given in Section 4. A numerical illustration of how the algorithmic results can be used for the analysis of the network consisting of 3 nodes, having 3 different service regimes, and an opportunity to simultaneously accommodate up to 40 users is presented in Section 5. Section 6 summarizes the results of the implemented study and briefly discusses possible directions for further research.

    Let us consider a SOQN that consists of K nodes. The illustration of the network structure is presented in Figure 1.

    Figure 1.  Structure of the network.

    Arrivals of users follow a marked Markov arrival process (MMAP). The irreducible underlying Markov chain {vt,t0} with the state space {1,,V} governs this process. The generator H of this chain is represented by the form

    H=Kk=0Hk

    where the diagonal entries of the matrix H0 are negative. The rates at which the chain {vt,t0} exits the appropriate states are determined by their moduli. The non-diagonal entries of the matrix H0 are nonnegative and specify the rates at which the chain {vt,t0} makes transitions without users arriving. The nonnegative entries in the matrix Hk specify the rates at which the chain {vt,t0} transits when a user arrives at the network's k-th node, k=¯1,K. The expression k=¯1,K indicates that the integer variable k takes values in the set {1,2,,K}.

    The average rate of arrivals to the k-th node of the network is calculated as λk=θHke,k=¯1,K, where the row vector θ defines the stationary probability distribution of the states of the chain {vt,t0}. The vector θ satisfies the equations θH=0,θe=1. Here, 0 represents the row vector with 0s, and e represents the column vector with 1s. The total rate of arrivals is defined as λ=Kk=1λk.

    Various details on the MMAP, its properties, and attributes, such as the distribution of moments of inter-arrival times, the coefficients of variation, and the correlation of inter-arrival times, can be found, for example, in [27,28,29,30,32].

    The network can process a maximum of N users simultaneously. When there are N users on the network, the user who arrives is regarded as lost. Every network's node functions as a queueing system having a single-server and a buffer with a capacity that is large enough to ensure that users are never lost.

    The network can serve users in L,L2, regimes. If the current service regime is l, the distribution of service time in the k-th node is exponential with intensity μ(l)k,l=¯1,L. After service completion at the k-th node, with probability pk,k the user transits for service to the k-th node, k=¯1,K,kk, or with probability pk,0 ends service in the network. Here, pk,k=0,Kk=0pk,k=1 for all k,k=¯1,K.

    The switching between the available service regimes is implemented by the following control strategy.

    Let two sets of integer numbers (thresholds), {L+1,L+2,,L+L1} and {L1,L2,,LL1}, such that

    0L1L+1<L2L+2<<LL1L+L1<N

    be fixed.

    We assume that the network instantly transitions to operating in the (l+1)-th regime if the number of users exceeds the threshold L+l and the current regime is l. For example, if the network operates in the first service regime, the number of users in the network is L+1, a new user is admitted to the network, and the service is immediately switched to the second regime.

    If some user leaves the network when it operates in the l-th regime, 2lL, and the number of users in the network drops to the value Ll1, the network immediately switches to operation in the (l1)-th regime.

    Let us present the sets listed below:

    W1={0,1,,L1},Wl={L+l1+1,L+l1+2,,Ll},l=¯2,L1,
    WL={L+L1+1,L+L1+2,,N},
    Fl={Ll+1,Ll+2,,L+l},l=¯1,L1.

    The network functions in the l-th regime if the quantity of users in the network equals n and nWl,l=¯1,L, as stated in the model description. If nFl,l=¯1,L1, the network can operate in both the l-th and (l+1)-th regime, and the used regime has to specified.

    The impatience of users (leaving the queue without receiving service because they waited in the buffer for too long) is an essential feature of many real-world systems. For a survey of the literature dealing with queues with impatient users, see, e.g., [41]. To take the impatience phenomenon into account, we assume the impatience of the users waiting in the buffers of the nodes of the network. A user in the k-th buffer reneges after an amount of time that is exponentially distributed with intensity βk,k=¯1,K, independently from other users who are waiting. Such a user permanently reneges from the network (is lost).

    We aim to analyze the invariant distribution of the network states and derive expressions for the calculation of the main performance characteristics of the network under the arbitrarily fixed values of the thresholds. As a result, a variety of optimization issues can be formulated and resolved.

    The continuous-time Markov chain

    ζt={nt,it,vt,m(1)t,,m(K)t},t0,

    can be used to define the behavior of the considered network.

    Here, during the instant t,

    ● the component nt defines the quantity of users in the network, nt=¯0,N;

    ● the component it is an indicator of the current regime of the network operation. It is defined only for the values nt such that ntFl,l=¯1,L1. It admits values 0 or 1. Namely, the value it=0 indicates the operation in the l-th regime, and it=1 indicates the operation in the (l+1)-th regime;

    ● the component vt specifies the status of the underlying process of the MMAP, vt=¯1,V;

    ● the component m(k)t defines the quantity of users in the k-th node, m(k)t=¯0,nt, Kk=1m(k)t=nt, k=¯1,K.

    Let us call the set {nt,it,vt,m(1)t,,m(K)t} of the states of Markov chain ζt,t0 enumerated in the reverse lexicographic order of the processes m(1)t,,m(K)t and the direct lexicographic order of the processes (it,vt) as level nt.

    The Markov chain {ζt,t0} is regular and irreducible. Its state space is finite. Therefore, the invariant probabilities of the states of this chain

    π(n,i,v,m(1),,m(K))=limtP{nt=n,it=i,vt=v,m(1)t=m(1),,m(K)t=m(K)}

    exist for all possible values of the queuing network's parameters.

    The probability of the states that correspond to the level n,n=¯0,N, can be formed as row vectors πn. The probability vectors πn,n=¯0,N, can be computed as the solution of the system of linear algebraic equations (known as Chapman-Kolmogorov, balance, or equilibrium equations):

    (π0,π1,,πN)Q=0,(π0,π1,,πN)e=1 (1)

    where the matrix Q represents the generator of the Markov chain {ζt,t0}.

    To calculate the vectors πn,n=¯0,N, the exact expression for the generator Q must be obtained.

    Let us present the following notation for future use in this paper:

    O is a zero matrix of the suitable size, and I is the identity matrix. If the size is not clear from context, it is indicated by a suffix, e.g., IK is the identity matrix of size K;

    ● the symbols and represent the Kronecker product and sum of matrices, respectively; refer to [42];

    b(k) is a vector of size K defined as b(k)={0,0,,0,1k,0,,0},k=¯1,K;

    diag{} denotes the diagonal matrix with the diagonal entries shown in brackets;

    Ω(l),l=¯1,L, is the square matrix of dimension K defined as:

    Ω(l)=diag{μ(l)k,k=¯1,K}(I+P),l=¯1,L,

    where the matrix P with the entries pk,k,k,k=¯1,K, defines the one-step transition probabilities of a user within the network;

    p(l),l=¯1,L, is the column vector defined as p(l)=(p(l)1,p(l)2,,p(l)K)T where p(l)k=pk,0μ(l)k,k=¯1,K;

    β is the column vector defined as β=(β1,β2,,βK)T.

    Let us first introduce the set of matrices that describe the behaviour of the K-dimensional stochastic process mt={m(1)t,,m(K)t},t0, defining the quantity of users in each node of the network. To define the transition rates of this process, conditioned on the knowledge that n users are present in the network working in the l-th regime at instant t, we need the following matrices:

    a) the matrix Tn(Ω(l)) establishes the process mt,t0, of transition intensities at the precise instant when a user ends the service in one network's node and moves on to another node, n=¯1,N,l=¯1,L.

    The following six steps make up the procedure for calculating the matrices Tn(Ω(l)),n=¯1,N,l=¯1,L:

    (1) Determine the matrices Ω(l)k,k=¯1,K2, that result from subtracting the K2k first rows and columns from the matrix Ω(l).

    (2) Calculate the set of matrices X(r,l)n,k using the recursive formulas:

    X(0,l)n,k=ωk,lr(l)k,1,n=¯1,N,k=¯1,K2,
    X(r,l)n,k=(ωk,lr(l)kr,1IOOX(r1,l)1,kωk,lr(l)kr,1IOOX(r1,l)2,kOOOωk,lr(l)kr,1IOOX(r1,l)n,k),
    r=¯1,r(l)k2,n=¯1,N,k=¯1,K2,l=¯1,L,

    where ωk,li1,i2 is the (i1,i2)th entry of the matrix Ω(l)k, and r(l)k represents how many rows there are in the matrix Ω(l)k.

    (3) Utilizing the recursive formulas, determine the set of the matrices Y(r,l)n,k:

    Y(0,l)n,k=ωk,l1,r(l)k,n=¯0,N1,k=¯1,K2,
    Y(r,l)n,k=(ωk,l1,r(l)krIY(r1,l)0,kOOOOωk,l1,r(l)krIY(r1,l)1,kOOOOOωk,l1,r(l)krIY(r1,l)n,k),
    r=¯1,r(l)k2,n=¯0,N1,k=¯1,K2,l=¯1,L.

    (4) Calculate the matrices X(l)n,k=X(rk2,l)n,k,n=¯1,N, and Y(l)n,k=Y(rk2,l)n,k, n=¯0,N1,k=¯1,K2,l=¯1,L.

    (5) Calculate the matrices T(k,l)n=T(k,l)n(Ω(l)) using the recursive formulas:

    T(0,l)n=(OΩ(l)K1,KOOOΩ(l)K,K1OΩ(l)K1,KOOOΩ(l)K,K1OOOOOOOΩ(l)K1,KOOOΩ(l)K,K1O),
    n=¯1,N,l=¯1,L,
    T(k,l)n=(OY(l)0,kOOOX(l)1,kT(k1,l)1Y(l)1,kOOOX(l)2,kT(k1,l)2OOOOOT(k1,l)n1Y(l)n1,kOOOX(l)n,kT(k1,l)n),
    n=¯1,N,k=¯1,K2,l=¯1,L.

    (6) Calculate the matrices Tn(Ω(l)) as

    Tn(Ω(l))=T(K2,l)n,n=¯1,N.

    b) The matrix Sn(p(l)) defines the intensities of the process mt,t0, of transitions that occur at the moment of a user service completion in some node and its departure from the network, n=¯1,N,l=¯1,L.

    The matrices Sn(p(l)), n=¯1,N,l=¯1,L, can be found as

    Sn(p(l))=S(K1,l)n(p(l)),n=¯1,N,l=¯1,L,

    where the matrices S(K1,l)n(p(l)) are recursively computed as:

    S(0,l)n(p(l))=p(l)K,
    S(k,l)n(p(l))=(p(l)KkIOOS(k1,l)1(p(l))p(l)KkIOOS(k1,l)2(p(l))OOOp(l)KkIOOS(k1,l)n(p(l))),
    k=¯1,K1,n=¯1,N,l=¯1,L.

    c) The matrix In(β) defines the intensities of the process mt,t0, of transitions that happen at the moment of some user loss due to impatience, n=¯2,N. The matrices In(β), n=¯2,N, can be found as In(β)=I(K1)n(β),n=¯2,N, where the matrices I(K1)n(β),n=¯2,N, are recursively obtained as

    I(0)n(β)=max{0,n1}βK,
    I(k)n(β)=(max{0,n1}βKkIOOI(k1)1(β)max{0,n2}βKkIOOI(k1)2(β)OOOmax{0,0}βKkIOOI(k1)n(β)),
    k=¯1,K1,n=¯2,N.

    Hereinafter, we assume that I1(β) is a zero matrix.

    d) The probabilities of the process mt transition at the time of a type-k user admittance to the network are defined by the matrix Pn(b(k)), k=¯1,K,n=¯0,N1. The paper [43] contains the algorithm for computing matrices Pn(b(r)). Using the denotations of our paper, we represent that algorithm below for the benefit of the reader. The matrices Pn(b),n=¯0,N1, where b=(b1,,bK),b{b(1),b(2),,b(K)}, can be computed as

    P0(b)=b,Pn(b)=P(K2)n,n=¯1,N1,

    where the matrices P(k)n of size (n+1)×(n+2),n=¯1,N1, are defined as

    P(0)n=(bK1bK0000bK1bK00000bK1bK),
    P(k)n=(bKk1˜b(k)00000TbKk1IP(k1)1OOO0TObKk1IP(k1)2OO0TOOObKk1IP(k1)n),k=¯1,K2,

    Here, the vectors ˜b(k) are defined as

    ˜b(k)=(bKk,bKk+1,,bK),k=¯1,K2.

    e) The overall rate at which the process mt exits the respective state is given by the moduli of the diagonal components of the diagonal matrix E(l)n, n=¯1,N,l=¯1,L. The matrices E(l)n are given by the formula

    E(l)n=diag{Tn(Ω(l))e+Sn(p(l))e+In(β)e},n=¯1,N.

    The presented formulas and algorithms used for the description of transition rates and probabilities of the process mt are based on the ideas presented in [40] for the simultaneous description of the phase-type service processes in several independent servers.

    Remark 1. A fact worth mentioning is that here, we compute the superfluous number of matrices T(l)n(Ω(l)), E(l)n, and S(l)n(p(l)) because if the network operates in the l-th regime, then the number n of users in the network should admit not an arbitrary value in the range from 0 to N but should belong to the interval [0,L+1], if l=1, [Ll1+1,L+l], if l=¯2,L1, and [LL1+1,N], if l=L. However, because we intend to build up the surfaces illustrating certain dependencies of the network performance indicators on the thresholds and solve the optimization problem, we would like to compute the generator Q of the Markov chain ζt,t0, for all possible sets of the thresholds L+l and Ll,l=¯1,L1. Therefore, if we compute all the matrices T(l)n(Ω(l)), E(l)n, and S(l)n(p(l)) for every value of n in the range from 0 to N and all values of l in the range from 1 to L from the early beginning, we will avoid redundant repeated computations during the building of the surfaces and the solution of the optimization problem. If it is required to make calculations only for the fixed set of the thresholds, computation of these matrices can be implemented only for the values on N from the corresponding diapason.

    After calculating the matrices mentioned above, which completely characterize the service process of users mt,t0, in all nodes of the network, we are ready to formulate the following statement.

    Theorem 1. The infinitesimal generator Q of the Markov chain ζt,t0, under study has a block-tridiagonal structure.

    The diagonal blocks Qn,n,n=¯0,N, of the generator are given by formulas:

    Q0,0=H0,
    Qn,n=H0(Tn(Ω(l))+E(l)n),forn0,nN,nWl,l=¯1,L,
    Qn,n=(H0(Tn(Ω(l))+E(l)n)OOH0(Tn(Ω(l+1))+E(l+1)n)),fornFl,l=¯1,L1,
    QN,N=H(TN(Ω(L))+E(L)N).

    The subdiagonal blocks Qn,n1,n=¯1,N, are given by formulas:

    Qn,n1=(OIV(Sn(p(l+1))+In(β))),forn=L+l+1Ll+1,l=¯1,L1.
    Qn,n1=IV(Sn(p(l+1))+In(β)),forn=L+l+1=Ll+1,l=¯1,L1.
    Qn,n1=IV(Sn(p(l))+In(β)),nWl,l=¯1,L,ifnL+l+1,l=¯1,L1.
    Qn,n1=(IV(Sn(p(l))+In(β))OOIV(Sn(p(l+1))+In(β))),
    nFl,fornLl+1,l=¯1,L1,
    Qn,n1=(IV(Sn(p(l))+In(β))IV(Sn(p(l+1))+In(β))),forn=Ll+1L+l+1,l=¯1,L1.

    The updiagonal blocks Qn,n+1,n=¯0,N1, are given by formulas:

    Qn,n+1=Kk=1HkPn(bk),nWl,fornLl,l=¯1,L,orn=Ll=L+l,l=¯1,L1,
    Qn,n+1=(Kk=1HkPn(bk)O),forn=LlL+l,l=¯1,L1,
    Qn,n+1=(Kk=1HkPn(bk)OOKk=1HkPn(bk)),fornFl,nL+l,l=¯1,L1,
    Qn,n+1=(Kk=1HkPn(bk)Kk=1HkPn(bk)),forn=L+lLl,l=¯1,L1.

    Proof. To implement the theorem's proof, every potential transition of the Markov chain ζt over an infinitesimally short interval is analyzed, and the transition intensities are rewritten in block matrix form. The simultaneous transition rates of two independent Markov chains, vt and mt, are defined via the Kronecker product of matrices.

    The meaning of the blocks of the generator is transparent taking into account the described above probabilistic meaning of the matrices H0,H1 and T(l)n(Ω(l)), E(l)n, and S(l)n(p(l)). The off-diagonal components of the diagonal blocks Qn,n define the rates of transition of the Markov chain ζt inside the level n. The modules of the negative diagonal entries of the blocks Qn,n define the rates of the departure of the Markov chain ζt from the states that belong to the level n. The entries of the blocks Qn,n1 define transition intensities of the Markov chain ζt from the level n to the level n1. Such transitions can occur during service completion of some user in the network or user loss due to impatience. The entries of the blocks Qn,n+1 define transition intensities of the Markov chain ζt from the level n to the level n+1. Such transitions can occur at the moments of a new user arrival and admission. Different sizes of some blocks are explained by the possibility of transitions of the number n of a level between the sets Wl and Fl, which imply the change (between one and two) of the cardinality of the state space of the component it.

    Remark 2. The number of blocks in the matrix Q depends on the maximum quantity N of users in the network but does not depend on the number L of the available service regimes. But the size of these blocks may depend on L and the relations between the thresholds L+l and Ll,l=¯1,L1. This size is minimal when the service rate control strategy is of the threshold, but not the hysteresis, type, i.e., L+l=Ll,l=¯1,L1. The square matrix Q may have a huge size. Even in the simpler case of the threshold strategy, it is equal to VNn=0Jn where Jm is the quantity of variants to distribute m users among K existing nodes, which is defined by the formula

    Jm=(m+K1K1)=(m+K1)!m!(K1)!,m=¯1,N,J0=1. (2)

    Therefore, certain efficient algorithms that use the generator's sparse structure are required to solve the system (1). Specifically, the approach from [44] can be suggested to determine the queueing network's and Markov chain's stationary probability distribution.

    Following the computation of the probability vectors πn,n=¯0,N, we may compute various performance metrics of the queuing network under study.

    The mean number of users in the network is given by the following expression

    Nnetwork=Nn=1nπne.

    The intensity of the output flow of successfully serviced users in the network can be found as

    λout=Ll=1nWl/{0}πn(IVSn(p(l)))e+L1l=1nFl1i=0π(n,i)(IVSn(p(l+i)))e.

    The output rate of users successfully serviced in the network from the k-th node is equal to

    λoutk=Ll=1nWl/{0}πn(IVSn(p(l,k)))e+L1l=1nFl1i=0π(n,i)(IVSn(p(l+i,k)))e,k=¯1,K,

    where p(l,k) is a column vector of dimension K that has all zero entries except the k-th entry (p(l,k))k, which is equal to p(l)k. The matrices Sn(p(l,k)) can be found using the same algorithm as for the matrices Sn(p(l)).

    The average quantity of users in the k-th node, k=¯1,K, is given by the following expression

    Nnodek=Ll=1nWl/{0}πn(IVJn(b(k)))e+L1l=1nFl1i=0π(n,i)(IVJn(b(k)))e

    where the matrices Jn(b),n=¯1,N, for the vectors b=(b1,,bK), which take values from the set b{b(1),b(2),,b(K)}, can be found as

    Jn(b)=J(K1)n(b),n=¯1,N,

    with the matrices J(K1)n(b),n=¯1,N, recursively obtained as

    J(0)n(b)=nbK,
    J(k)n(b)=(nbKkIOOJ(k1)1(b)(n1)bKkIOOJ(k1)2(b)OOObKkIOOJ(k1)n(b)),k=¯1,K1,n=¯1,N.

    The average quantity of busy servers in the k-th node, k=¯1,K, is obtained using the formula

    Nservk=Ll=1nWl/{0}πn(IVSn(b(k)))e+L1l=1nFl1i=0π(n,i)(IVSn(b(k)))e.

    The average quantity of busy servers in the network is given by the following expression

    Nserv=Kk=1Nservk.

    The average quantity of users in the k-th node's buffer, k=¯1,K, is given by the following expression

    Nbufk=Ll=1nWl/{0}πn(IVIn(b(k)))e+L1l=1nFl1i=0π(n,i)(IVIn(b(k)))e=N(k)nodeN(k)serv.

    The average quantity of users waiting in all buffers of the network can be calculated using the formula

    Nbuf=Kk=1Nbufk.

    The probability that the network operates in the l-th regime at an arbitrary epoch is determined by the following expression

    Pregimel=nWlπne+nFl,lLπ(n,0)e+nFl1,l1π(n,1)e,l=¯1,L.

    The intensity of the regime increase is given by the formula

    ϕ+=L1l=1π(L+l,0)Kk=1HkIJL+l.

    The intensity of the regime decrease is given by the following expression

    ϕ=L1l=1Φl=ϕ+

    where

    Φl={π(Ll+1,1)(IV(ILl+1(β)+SLl+1(p(l+1))) if LlL+l,πLl+1(IV(ILl+1(β)+SLl+1(p(l+1))) if Ll=L+l.

    The average intensity of regime switching is equal to

    ϕ=ϕ++ϕ=2ϕ+=2ϕ.

    The probability of an arbitrary user loss upon arrival caused by the residence of N users in the network is calculated by the expression

    Pentloss=λ1πN((HH0)IJN)e.

    The probability of an arbitrary type-k user loss upon arrival due to the residence of N users in the network is computed as

    Pentlossk=λ1kπN(HkIJN)e,k=¯1,K.

    The probability of an arbitrary user loss upon arrival at the k-th node due to the residence of N users in the network is given by the following expression

    Pentlossarbk=λ1πN(HkIJN)e,k=¯1,K.

    The probability of an arbitrary user loss due to impatience is equal to

    Pimploss=λ1Kk=1Nbufkβk=λ1(Ll=1nWl/{0}πn(IVIn(β))e+L1l=1nFl1i=0π(n,i)(IVIn(β))e).

    The probability of an arbitrary user loss due to impatience in the k-th node is computed as

    Pimplossk=λ1Nbufkβk=λ1(Ll=1nWl/{0}πn(IVIn(β(k)))e+L1l=1nFl1i=0π(n,i)(IVIn(β(k)))e),k=¯1,K,

    where β(k) is a column vector of size K with all zero entries except the k-th entry (β(k))k, which is equal to βk.

    The probability of an arbitrary user loss is given by the following expression

    Ploss=Pentloss+Pimploss=1λoutλ.

    Controlling the accuracy of the calculation of the stationary distribution of the network states is made easier by the existence of two distinct expressions for computing the probability Ploss and the average intensity of regime switching.

    An arbitrary user's loss probability in the k-th node is determined as

    Plossk=Pentlossarbk+Pimplossk,k=¯1,K.

    An arbitrary user's probability of receiving successful service within the network is determined by the formula

    Psucc=1Ploss=λoutλ.

    Let us show the numerical example that verifies the viability of the suggested methods and formulas, and partially highlights the impact of variation of the thresholds on the value of the key performance indicators of the system and the potential to apply the outcomes to managerial objectives.

    In this example, we examine a queueing network with K=3 nodes.

    The MMAP flow of users arriving at the network is determined by the following matrices:

    H0=(9.30.30.32.7),H1=(3.30.030.0090.579),
    H2=(2.40.150.0121.2),H3=(3.060.0600.6).

    The average arrival rate for this arrival flow is λ=4.8606. The average arrival rate λk and the coefficients of variation c(k)var and correlation c(k)cor of successive inter-arrival times to the k-th node, k=1,2,3, have the following values:

    λ1=1.6103,c(1)var=1.77393c(1)cor=0.181652,
    λ2=1.7108,c(2)var=2.05727c(2)cor=0.148899,
    λ3=1.5395,c(3)var=1.16264c(3)cor=0.0462668.

    We assume that there are L=3 possible service regimes of the network operation. Under the first regime, the service times in the nodes are exponentially distributed with parameters

    μ(1)1=1.5,μ(1)2=1,μ(1)3=0.9,

    respectively. Under the second regime, the parameters of the exponential distribution of the service time are

    μ(2)1=2μ(1)1,μ(2)2=2μ(1)2,μ(2)3=2μ(1)3,

    and under the third regime, the parameters are

    μ(3)1=3μ(1)1,μ(3)2=3μ(1)2,μ(3)3=3μ(1)3.

    The transition probabilities of users after the service completion in the nodes are defined as

    p1,0=3/5,p1,2=2/15,p1,3=4/15,p2,0=0.7,
    p2,1=0.1,p2,3=0.2,p3,0=2/3,p3,1=2/9,p3,2=1/9.

    The rates of the users' departure from the buffers of the nodes due to impatience are defined as follows:

    β1=0.01,β2=0.02,β3=0.015.

    In this numerical example, we assume that up to N=40 users can obtain service in the network at the same time. The regime switching is defined by the four parameters L1,L2,L+1 and L+2.

    The purpose of the numerical example is to show the dependence of the main network's performance measures on the switching parameters. However, there is no opportunity to build 5 D figures. Therefore, for better visualization of the results, let us fix the thresholds defining the rule of the switching between the first and the second regimes as L1=5 and L+1=10 and vary the thresholds L2 and L+2, defining the rule of the switching between the second and third regimes as follows: The threshold L+2 varies in the interval [L+1+1,N), and the threshold L2 varies over the interval [L+1+1,L+2] with the same step 1.

    Figures 2 and 3 illustrate the dependence of the average quantity Nnetwork and the average total number Nbuf of users in buffers on the parameters L2 and L+2.

    Figure 2.  Nnetwork as function of L2 and L+2.
    Figure 3.  Nbuf as function of L2 and L+2.

    Complementary to Figure 2, the dynamics of Nnetwork are illustrated in Table 1 where values of Nnetwork are presented for values of L2 and L+2 in some smaller range.

    Table 1.  Values of Nnetwork for different values of L2 and L+2.
    11 12 13 14 15 16 17 18 19 20
    11 19.089
    12 19.256 19.451
    13 19.422 19.627 19.834
    14 19.587 19.801 20.019 20.237
    15 19.753 19.973 20.199 20.428 20.655
    16 19.921 20.146 20.378 20.615 20.853 21.088
    17 20.090 20.319 20.556 20.799 21.044 21.290 21.532
    18 20.263 20.494 20.735 20.981 21.233 21.486 21.738 21.986
    19 20.438 20.673 20.915 21.164 21.419 21.678 21.938 22.195 22.447
    20 20.617 20.853 21.097 21.349 21.606 21.868 22.132 22.397 22.659 22.914
    21 20.798 21.037 21.283 21.535 21.794 22.057 22.325 22.594 22.863 23.128
    22 20.982 21.222 21.470 21.724 21.983 22.248 22.516 22.788 23.061 23.333
    23 21.168 21.410 21.659 21.914 22.174 22.439 22.708 22.981 23.256 23.532
    24 21.355 21.599 21.850 22.106 22.367 22.632 22.902 23.174 23.450 23.727
    25 21.543 21.790 22.042 22.299 22.561 22.827 23.096 23.368 23.643 23.921
    26 21.733 21.981 22.235 22.493 22.756 23.022 23.291 23.563 23.838 24.114
    27 21.923 22.172 22.428 22.688 22.951 23.218 23.487 23.759 24.033 24.308
    28 22.112 22.364 22.621 22.882 23.146 23.414 23.683 23.955 24.228 24.502
    29 22.302 22.555 22.813 23.075 23.3413 23.609 23.878 24.150 24.422 24.696
    30 22.490 22.745 23.005 23.268 23.534 23.802 24.073 24.344 24.616 24.889
    31 22.677 22.934 23.194 23.459 23.726 23.995 24.265 24.536 24.808 25.08
    32 22.863 23.120 23.382 23.647 23.915 24.184 24.455 24.726 24.997 25.269
    33 23.046 23.304 23.567 23.833 24.101 24.371 24.642 24.913 25.184 25.455
    34 23.226 23.485 23.749 24.016 24.284 24.554 24.825 25.096 25.367 25.637
    35 23.402 23.663 23.927 24.194 24.463 24.733 25.004 25.275 25.545 25.814
    36 23.574 23.835 24.101 24.368 24.637 24.907 25.178 25.448 25.718 25.986
    37 23.741 24.003 24.268 24.535 24.805 25.075 25.345 25.615 25.884 26.152
    38 23.901 24.163 24.429 24.697 24.966 25.235 25.505 25.774 26.042 26.309
    39 24.054 24.317 24.582 24.85 25.118 25.387 25.656 25.925 26.192 26.457

     | Show Table
    DownLoad: CSV

    It is seen from these figures that the minimal values of Nnetwork and Nbuf are achieved for small values of the thresholds L2 and L+2. When these thresholds increase (this means that the third regime is used only for a larger number of users in the network), the values of Nnetwork and Nbuf increase quite sharply.

    This is easily understandable, as the service rate during the use of the third regime is three times higher than during the use of the first regime and is 1.5 times higher than during the use of the second regime.

    Figures 46 illustrate the dependence of the probabilities Pregimel on the fact that, at any arbitrary moment, the network operates in the l-th regime, l=1,2,3, on the parameters L2 and L+2.

    Figure 4.  Pregime1 as function of L2 and L+2.
    Figure 5.  Pregime2 as function of L2 and L+2.
    Figure 6.  Pregime3 as function of L2 and L+2.

    The maximum value of Pregime1 is achieved for small values of L2 and L+2 because such small values imply a more frequent use of the fastest, the third, regime of operation (this is confirmed by Figure 6) and higher chances that the number of users in the network will drop below the value L1+1 and, therefore, the first regime will be used. The maximum value of Pregime2 is achieved for large values of L2 and L+2 because the second regime is used until the number of users in the network drops below the value L2+1, which implies the longer use of the second regime.

    Figure 7 illustrates the dependence of the average intensity ϕ of regime switching on the parameters L2 and L+2.

    Figure 7.  ϕ as function of L2 and L+2.

    Figures 811 illustrate the dependence of the probabilities Pentloss of a user loss at the entrance to the network and the probabilities Pentlossarbl of a user loss at the entrance to the l-th node of the network, l=1,2,3, on the parameters L2 and L+2.

    Figure 8.  Pentloss as function of L2 and L+2.
    Figure 9.  Pentlossarb1 as function of L2 and L+2.
    Figure 10.  Pentlossarb2 as function of L2 and L+2.
    Figure 11.  Pentlossarb3 as function of L2 and L+2.

    Figures 1215 illustrate the dependence of the probabilities Pimploss of a user loss due to impatience in the network and the probabilities Pimplossl of a user loss due to impatience from the buffer of the l-th node of the network, l=1,2,3, on the parameters L2 and L+2.

    Figure 12.  Pimploss as function of L2 and L+2.
    Figure 13.  Pimploss1 as function of L2 and L+2.
    Figure 14.  Pimploss2 as function of L2 and L+2.
    Figure 15.  Pimploss3 as function of L2 and L+2.

    Figure 16 illustrates the dependence of the loss probability Ploss of an arbitrary user (due to all reasons) on the parameters L2 and L+2.

    Figure 16.  Ploss as function of L2 and L+2.

    Complementary to Figure 16, the dynamics of Ploss are illustrated in Table 2.

    Table 2.  Values of Ploss for different values of L2 and L+2.
    11 12 13 14 15 16 17 18 19 20
    11 0.0788
    12 0.0797 0.0807
    13 0.0806 0.0817 0.0828
    14 0.0815 0.0826 0.0838 0.0850
    15 0.0824 0.0836 0.0848 0.0861 0.0873
    16 0.0833 0.0845 0.0858 0.0871 0.0885 0.0899
    17 0.0843 0.0855 0.0869 0.0882 0.0896 0.0911 0.0925
    18 0.0853 0.0866 0.0879 0.0893 0.0908 0.0923 0.0938 0.0953
    19 0.0863 0.0876 0.0890 0.0905 0.0920 0.0935 0.0951 0.0967 0.0983
    20 0.0874 0.0888 0.0902 0.0916 0.0932 0.0947 0.0964 0.0981 0.0998 0.1015
    21 0.0886 0.0899 0.0913 0.0928 0.0944 0.0960 0.0977 0.0994 0.1012 0.1030
    22 0.0897 0.0911 0.0926 0.0941 0.0957 0.0973 0.0990 0.1008 0.1026 0.1045
    23 0.0909 0.0924 0.0939 0.0954 0.0970 0.0987 0.1004 0.1022 0.1041 0.1060
    24 0.0922 0.093 0.0952 0.0968 0.0984 0.1001 0.1019 0.1037 0.1056 0.1075
    25 0.0935 0.0950 0.0966 0.0982 0.0998 0.1016 0.1034 0.1052 0.1071 0.1091
    26 0.0949 0.0964 0.0980 0.0996 0.1013 0.1031 0.1049 0.1068 0.1088 0.1108
    27 0.0963 0.0979 0.0995 0.1012 0.1029 0.1047 0.1066 0.1085 0.1105 0.1125
    28 0.0978 0.0994 0.1011 0.1028 0.1045 0.1064 0.1083 0.1102 0.1122 0.1143
    29 0.0994 0.1010 0.1027 0.1044 0.1062 0.1081 0.1100 0.1120 0.1141 0.1162
    30 0.1010 0.1027 0.1044 0.1062 0.1080 0.1099 0.1119 0.1139 0.1160 0.1182
    31 0.1027 0.1044 0.1062 0.1080 0.1099 0.1118 0.1139 0.1159 0.1181 0.1203
    32 0.1045 0.1062 0.1080 0.1099 0.1118 0.1138 0.1159 0.1180 0.1202 0.1225
    33 0.1063 0.1081 0.1100 0.1119 0.1139 0.1159 0.1180 0.1202 0.1225 0.1248
    34 0.1083 0.1101 0.1120 0.1140 0.1160 0.1181 0.1203 0.1225 0.1248 0.1272
    35 0.1104 0.1122 0.1142 0.1162 0.1183 0.1205 0.1227 0.1250 0.1274 0.1298
    36 0.1125 0.1145 0.1165 0.1186 0.1207 0.1229 0.1252 0.1276 0.1300 0.1325
    37 0.1148 0.1168 0.1189 0.1211 0.1233 0.1256 0.1279 0.1303 0.1328 0.1354
    38 0.1173 0.1194 0.1215 0.1237 0.1260 0.1283 0.1308 0.1333 0.1359 0.1385
    39 0.1199 0.1220 0.1242 0.1265 0.1289 0.1313 0.1338 0.1364 0.1391 0.1418

     | Show Table
    DownLoad: CSV

    Because we have fixed L+1=10, and the thresholds L2 and L+2 have to satisfy the inequalities L+1<L2L+2<N, the possible values of the thresholds L2 and L+2 range from 11 to 39. The loss probability Ploss reaches its minimum value of 0.07887 when L2=L+2=11. When L2=L+2=39, the maximum value of the loss probability Ploss is reached and equals 0.23454. This fact is obvious because when L2=L+2=11, the network starts operation in the fastest, the third, service regime as soon as possible. When L2=L+2=39, the third regime is switched on only when the number of users in the system is equal to the maximum admissible value of 40. Therefore, many users are lost at the entrance to the network and due to impatience. However, when deciding on the selection of the values of the thresholds, it is necessary to take into account that the use of faster service regimes by default is more costly compared to slower service regimes. Also, it is undesirable to frequently change service regimes because such a change in a real-world network can require some expenditures related to switching to another equipment or inviting or dismissing staff.

    Therefore, optimization of the operation of the network requires an exact definition of the cost criterion. Let us assume that the following cost criterion is used to define the network's operational quality:

    E=E(L2,L+2)=aλoutbλPentlosscλPimplossLl=1elPregimeldϕ. (3)

    Here, a is a profit obtained by the system via the service of one user, b is a penalty paid by the network for one user loss upon arrival, c is a penalty paid by the network for one user loss due to impatience, el,l=¯1,L, is the cost of maintaining the l-th operation regime per unit time, and d is a charge paid by the network for one switch of an operation regime.

    Thus, the cost criterion E represents the average network's revenue per unit of time. Our aim is to find the values of the thresholds L2 and L+2 providing the maximum to the function E(L2,L+2).

    The cost coefficients in this numerical example are fixed at the following values:

    a=3,b=3,c=6,e1=1,e2=2,e3=8,d=0.5.

    Figure 17 shows how the thresholds L2 and L+2 affect the cost criteria E.

    Figure 17.  E as function of L2 and L+2.

    When L+2=20 and L2=15, the cost criterion reaches its optimal value of E=5.19909. Thus, to obtain the maximal revenue under the fixed above values of the network parameters, it is necessary to switch from the second to the third service regime when the number of users in the network becomes equal to (L+2)+1=21 and switch back to the second service regime when the number of users drops to L2=15.

    All figures presented above illustrate the dependence of the cost criterion E on the thresholds L2 and L+2 under the fixed values of the thresholds L1=5 and L+1=10. Let us now assume that L1=L+1=L1 and L2=L+2=L2, i.e., the hysteresis strategy turns into the threshold strategy. This means that if the number nt of users in the network does not exceed L1, then the network operates in the first regime. If the number nt belongs to the interval (L1+1,L2], then the network operates in the second regime. If the number nt exceeds L2, then the network operates in the third regime.

    Figures 18 and 19 illustrate the dependence of the loss probabilities Pentloss of an arbitrary user upon arrival and Pimploss of an arbitrary user due to impatience on the parameters L1 and L2.

    Figure 18.  Pentloss as function of L1 and L2.
    Figure 19.  Pimploss as function of L1 and L2.

    Figure 20 illustrates the dependence of the cost criterion E(L1,L2) on the parameters L1 and L2.

    Figure 20.  The cost criterion E(L1,L2) as function of L1 and L2.

    The optimal values of the thresholds are as follows: L1=0 and L2=15. This means that the network uses the first regime only when it is empty; once a user arrives, it is serviced in the second regime. When the number of users in the network reaches the value of 16, the network starts operation in the third regime and maintains this regime until the number of users in the network drops to the value of 15. The maximal value of the cost criterion is 5.13969.

    Let us return to the hysteresis-type control by the network operation. A numerical solution to the problem of determining the optimal value of the cost criterion under the network's fixed parameters can be found using the results obtained for the computation of the stationary distribution of the network states, the primary performance characteristics, and the cost criterion value. In our example, the value of the cost criterion is the function of four thresholds, L1,L+1,L2, and L+2. It can be computed that the maximum value of the cost criterion is equal to 5.31252 and is achieved for the following values of the thresholds: L1=0,L+1=2,L2=13,L+2=18. The achieved value of 5.31252 of the cost criterion is higher than the optimal value of 5.13969 of the criterion under the use of the optimal threshold strategy due to the more seldom switching of regimes and the presence of a charge for the switch of the regimes in the cost criterion.

    In a more general case, when the number L of available regimes is higher than two (and the number of the thresholds is 2L), the problem of finding the maximal value of the criterion and the optimal values of the corresponding thresholds can be deeply complicated by the existence of a huge number of possible combinations of the threshold values. Therefore, this problem deserves a separate consideration. The use of some derivative-free methods of optimization, see, e.g., [45,46], can be recommended. As mentioned above, the value of our results consists in providing the possibility to exactly compute the value of the cost criterion for any fixed set of control strategies during the implementation of the search for the optimal values of the thresholds.

    It is worth noting that we assumed that the values of the service rates μ(l)k,k=¯1,K,l=¯1,L, in the nodes of the network under the fixed regimes of the network operation are fixed. In potential real-world applications, these values can also not be fixed but have to be chosen from some set. Our results can be used to optimize the selection of these rates' values and the corresponding threshold values.

    Remark 3. Described above computations were implemented using Wolfram Mathematica on a Lenovo notebook with an Intel(R) Core(TM) i7-1165G7 2.80GHz and 16 GB RAM. Running time for computation of the optimal value of the cost criterion E(L2,L+2) defined by formula (3) was equal to 8419 seconds, i.e., 18 seconds for one point (L2,L+2) (the total number of points is 465). Because this computation time was acceptable for preparation of the presented examples, no optimization of the code was made. Computation time can be significantly reduced via such an optimization and the use of a more powerful notebook or PC.

    A quite long computation time is explained by the large size of the generator Q. As mentioned above in the simplest case of the threshold strategy, this size is equal to VNn=0Jn, where the numbers Jn,n=¯0,N, are defined by formula (2). In the considered example of SOQN, where we assume K=3 nodes and admission of up to N=40 customers to the network simultaneously, the number JN is equal to 861. If we consider the network consisting of four nodes and decreased N to 15, we will have about the same (816) size of the block JN and a similar computation time.

    We considered a user loss SOQN with the bursty MMAP-type arrival process, single-server nodes with a controlled service regime, a hysteresis-type control policy, and impatient users. Under the fixed parameters of the control policy, the stationary behavior of this SOQN is described by a multidimensional Markov chain, whose components define the total number of users in the network, the used service regime (if it is not uniquely defined by the number of users in the network), the state of the underlying process of the MMAP, and the number of users in each node of the network. The generator of this chain is obtained as a block tridiagonal matrix. Formulas for computation of the key performance measures of the network are derived. Numerical illustrations of the algorithmic and analytical results obtained are provided.

    The results can be applied to managerial objectives, such as the selection of possible variants of service organization in the nodes, including the choice of the suitable equipment and the corresponding staff; routing of users in the network; pricing; and the optimal dynamical scheduling of the variants of service organization depending on the current load of the network.

    The results can be generalized into several directions, such as the consideration of back-ordering SOQNs having an infinite or finite buffer for storing users who did not succeed in entering the core network upon arrival because it was completely busy; an account of possible user's impatience during waiting in this buffer; the presence of an orbit for user retrials; or the arrival and impatience rate control. The results from [47] are planned to be used for implementing these generalizations.

    Ciro D'Apice: Conceptualization, Methodology, Validation, Investigation, Writing-original draft, Writing-review & editing, Project administration; Alexander Dudin: Conceptualization, Methodology, Formal analysis, Investigation, Writing-original draft, Writing-review & editing, Project administration; Sergei Dudin: Methodology, Software, Formal analysis, Investigation, Supervision, Writing-original draft; Rosanna Manzo: Conceptualization, Software, Validation, Formal analysis, Investigation, Writing-original draft, Writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Rosanna Manzo is the Guest Editor of special issue "Managing complex systems by simulation and optimization techniques" for AIMS Mathematics. Rosanna Manzo was not involved in the editorial review and the decision to publish this article.



    [1] F. Bray, J. Ferlay, I. Soerjomataram, R. L. Siegel, L. A. Torre, A. Jemal, Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA Cancer J. Clin., 68 (2018), 394–424. https://doi.org/10.3322/caac.21492 doi: 10.3322/caac.21492
    [2] R. Dent, M. Trudeau, K. I. Pritchard, W. M. Hanna, H. K. Kahn, C. A. Sawka, et al., Triple-negative breast cancer: clinical features and patterns of recurrence, Clin. Cancer Res., 13 (2007), 4429–4434. https://doi.org/10.1158/1078-0432.CCR-06-3045 doi: 10.1158/1078-0432.CCR-06-3045
    [3] A. Marra, G. Viale, G. Curigliano, Recent advances in triple negative breast cancer: the immunotherapy era, BMC Medicine, 17 (2019), 90. https://doi.org/10.1186/s12916-019-1326-5 doi: 10.1186/s12916-019-1326-5
    [4] G. Zarrilli, G. Businello, M. V. Dieci, S. Paccagnella, V. Carraro, R. Cappellesso, et al., The tumor microenvironment of primitive and metastatic breast cancer: implications for novel therapeutic strategies, Int. J. Mol. Sci., 21 (2020), 8102. https://doi.org/10.3390/ijms21218102 doi: 10.3390/ijms21218102
    [5] S. D. Soysal, A. Tzankov, S. E. Muenst, Role of the tumor microenvironment in breast cancer, Pathobiology, 82 (2015), 142–152. https://doi.org/10.1159/000430499 doi: 10.1159/000430499
    [6] P. V. Loo, T. Voet, Single cell analysis of cancer genomes, Curr. Opin. Genet. Dev., 24 (2014), 82–91. https://doi.org/10.1016/j.gde.2013.12.004 doi: 10.1016/j.gde.2013.12.004
    [7] A. A. Pollen, T. J. Nowakowski, J. Shuga, X. Wang, A. A. Leyrat, J. H. Lui, et al., Low-coverage single-cell mRNA sequencing reveals cellular heterogeneity and activated signaling pathways in developing cerebral cortex, Nat. Biotechnol., 32 (2014), 1053–1058. https://doi.org/10.1038/nbt.2967 doi: 10.1038/nbt.2967
    [8] B. Treutlein, D. G. Brownfield, A. R. Wu, N. F. Neff, G. L. Mantalas, F. H. Espinoza, et al., Reconstructing lineage hierarchies of the distal lung epithelium using single-cell RNA-seq, Nature, 509 (2014), 371–375. https://doi.org/10.1038/nature13173 doi: 10.1038/nature13173
    [9] Q. H. Nguyen, N. Pervolarakis, K. Blake, D. Ma, R. T. Davis, N. James, et al., Profiling human breast epithelial cells using single cell RNA sequencing identifies cell diversity, Nat. Commun., 9 (2018), 2028. https://doi.org/10.1038/s41467-018-04334-1 doi: 10.1038/s41467-018-04334-1
    [10] M. Bartoschek, N. Oskolkov, M. Bocci, J. Lövrot, C. Larsson, M. Sommarin, et al., Spatially and functionally distinct subclasses of breast cancer-associated fibroblasts revealed by single cell RNA sequencing, Nat. Commun., 9 (2018), 5150. https://doi.org/10.1038/s41467-018-07582-3 doi: 10.1038/s41467-018-07582-3
    [11] D. Hanahan, R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646–674. https://doi.org/10.1016/j.cell.2011.02.013 doi: 10.1016/j.cell.2011.02.013
    [12] B. Muz, P. de la Puente, F. Azab, A. K. Azab, The role of hypoxia in cancer progression, angiogenesis, metastasis, and resistance to therapy, Hypoxia (Auckl), 3 (2015), 83–92. https://doi.org/10.2147/HP.S93413 doi: 10.2147/HP.S93413
    [13] X. Jing, F. Yang, C. Shao, K. Wei, M. Xie, H. Shen, et al., Role of hypoxia in cancer therapy by regulating the tumor microenvironment, Mol. Cancer, 18 (2019), 157. https://doi.org/10.1186/s12943-019-1089-9 doi: 10.1186/s12943-019-1089-9
    [14] X. Sun, H. Luo, C. Han, Y. Zhang, C. Yan, Identification of a hypoxia-related molecular classification and hypoxic tumor microenvironment signature for predicting the prognosis of patients with triple-negative breast cancer, Front. Oncol., 11 (2021), 700062. https://doi.org/10.3389/fonc.2021.700062 doi: 10.3389/fonc.2021.700062
    [15] X. Yang, X. Weng, Y. Yang, M. Zhang, Y. Xiu, W. Peng, et al., A combined hypoxia and immune gene signature for predicting survival and risk stratification in triple-negative breast cancer, Aging (Albany NY), 13 (2021), 19486–19509. https://doi.org/10.18632/aging.203360 doi: 10.18632/aging.203360
    [16] R. Gao, S. Bai, Y. C. Henderson, Y. Lin, A. Schalck, Y. Yan, et al., Delineating copy number and clonal substructure in human tumors from single-cell transcriptomes, Nat. Biotechnol., 39 (2021), 599–608. https://doi.org/10.1038/s41587-020-00795-2 doi: 10.1038/s41587-020-00795-2
    [17] M. J. Goldman, B. Craft, M. Hastie, K. Repečka, F. McDade, A. Kamath, et al., Visualizing and interpreting cancer genomics data via the Xena platform, Nat. Biotechnol., 38 (2020), 675–678. https://doi.org/10.1038/s41587-020-0546-8 doi: 10.1038/s41587-020-0546-8
    [18] D. M. Gendoo, N. Ratanasirigulchai, M. S. Schroder, L. Paré, J. S. Parker, A. Prat, et al., Genefu: an R/Bioconductor package for computation of gene expression-based signatures in breast cancer, Bioinformatics, 32 (2016), 1097–1099. https://doi.org/10.1093/bioinformatics/btv693 doi: 10.1093/bioinformatics/btv693
    [19] P. Jezequel, D. Loussouarn, C. Guerin-Charbonnel, L. Campion, A. Vanier, W. Gouraud, et al., Gene-expression molecular subtyping of triple-negative breast cancer tumours: importance of immune response, Breast Cancer Res., 17 (2015), 43. https://doi.org/10.1186/s13058-015-0550-y doi: 10.1186/s13058-015-0550-y
    [20] Y. Hao, S. Hao, E. Andersen-Nissen, W. M. Mauck III, S. Zheng, A. Butler, et al., Integrated analysis of multimodal single-cell data, Cell, 184 (2021), 3573–3587. https://doi.org/10.1016/j.cell.2021.04.048 doi: 10.1016/j.cell.2021.04.048
    [21] D. Aran, A. P. Looney, L. Liu, E. Wu, V. Fong, A. Hsu, et al., Reference-based analysis of lung single-cell sequencing reveals a transitional profibrotic macrophage, Nat. Immunol., 20 (2019), 163–172. https://doi.org/10.1038/s41590-018-0276-y doi: 10.1038/s41590-018-0276-y
    [22] A. Liberzon, C. Birger, H. Thorvaldsdottir, M. Ghandi, J. P. Mesirov, P. Tamayo, The Molecular Signatures Database (MSigDB) hallmark gene set collection, Cell Syst., 1 (2015), 417–425. https://doi.org/10.1016/j.cels.2015.12.004 doi: 10.1016/j.cels.2015.12.004
    [23] S. Jin, C. F. Guerrero-Juarez, L. Zhang, I. Chang, R. Ramos, C. H. Kuan, et al., Inference and analysis of cell-cell communication using CellChat, Nat. Commun., 12 (2021), 1088. https://doi.org/10.1038/s41467-021-21246-9 doi: 10.1038/s41467-021-21246-9
    [24] J. F. Prud'homme, F. Fridlansky, M. Le Cunff, M. Atger, C. Mercier-Bodart, M. F. Pichon, et al., Cloning of a gene expressed in human breast cancer and regulated by estrogen in MCF-7 cells, DNA, 4 (1985), 11–21. https://doi.org/10.1089/dna.1985.4.11 doi: 10.1089/dna.1985.4.11
    [25] M. C. Rio, J. P. Bellocq, J. Y. Daniel, C. Tomasetto, R. Lathe, M. P. Chenard, et al., Breast cancer-associated pS2 protein: synthesis and secretion by normal stomach mucosa, Science, 241 (1988), 705–708. https://doi.org/10.1126/science.3041593 doi: 10.1126/science.3041593
    [26] T. Vogl, A. Stratis, V. Wixler, T. Völler, S. Thurainayagam, S. K. Jorch, et al., Autoinhibitory regulation of S100A8/S100A9 alarmin activity locally restricts sterile inflammation, J. Clin. Invest., 128 (2018), 1852–1866. https://doi.org/10.1172/JCI89867 doi: 10.1172/JCI89867
    [27] Q. Fang, S. Yao, G. Luo, X. Zhang, Identification of differentially expressed genes in human breast cancer cells induced by 4-hydroxyltamoxifen and elucidation of their pathophysiological relevance and mechanisms, Oncotarget, 9 (2018), 2475–2501. https://doi.org/10.18632/oncotarget.23504 doi: 10.18632/oncotarget.23504
    [28] N. O'Brien, T. M. Maguire, N. O'Donovan, N. Lynch, A. D. Hill, E. McDermott, et al., Mammaglobin a: a promising marker for breast cancer, Clin. Chem., 48 (2002), 1362–1364. https://doi.org/10.1093/clinchem/48.8.1362 doi: 10.1093/clinchem/48.8.1362
    [29] M. Zafrakas, B. Petschke, A. Donner, F. Fritzsche, G. Kristiansen, R. Knüchel, et al., Expression analysis of mammaglobin A (SCGB2A2) and lipophilin B (SCGB1D2) in more than 300 human tumors and matching normal tissues reveals their co-expression in gynecologic malignancies, BMC Cancer, 6 (2006), 88. https://doi.org/10.1186/1471-2407-6-88 doi: 10.1186/1471-2407-6-88
    [30] D. Carter, J. F. Douglass, C. D. Cornellison, M. W. Retter, J. C. Johnson, A. A. Bennington, et al., Purification and characterization of the mammaglobin/lipophilin B complex, a promising diagnostic marker for breast cancer, Biochemistry, 41 (2002), 6714–6722. https://doi.org/10.1021/bi0159884 doi: 10.1021/bi0159884
    [31] T. L. Colpitts, P. Billing-Medel, P. Friedman, E. N. Granados, M. Hayden, S. Hodges, et al., Mammaglobin is found in breast tissue as a complex with BU101, Biochemistry, 40 (2001), 11048–11059. https://doi.org/10.1021/bi010284f doi: 10.1021/bi010284f
    [32] S. Robson, S. Pelengaris, M. Khan, c-Myc and downstream targets in the pathogenesis and treatment of cancer, Recent Pat. Anticancer Drug Discov., 1 (2006), 305–326. https://doi.org/10.2174/157489206778776934 doi: 10.2174/157489206778776934
    [33] P. C. Fernandez, S. R. Frank, L. Wang, M. Schroeder, S. Liu, J. Greene, et al., Genomic targets of the human c-Myc protein, Genes Dev., 17 (2003), 1115–1129. https://doi.org/10.1101/gad.1067003 doi: 10.1101/gad.1067003
    [34] J. H. Patel, A. P. Loboda, M. K. Showe, L. C. Showe, S. B. McMahon, Analysis of genomic targets reveals complex functions of MYC, Nat. Rev. Cancer, 4 (2004), 562–568. https://doi.org/10.1038/nrc1393 doi: 10.1038/nrc1393
    [35] C. Attwooll, E. L. Denchi, K. Helin, The E2F family: specific functions and overlapping interests, EMBO J., 23 (2004), 4709–4716. https://doi.org/10.1038/sj.emboj.7600481 doi: 10.1038/sj.emboj.7600481
    [36] T. Yu, L. Liang, X. Zhao, Y. Yin, Structural and biochemical studies of the extracellular domain of Myelin protein zero-like protein 1, Biochem. Biophys. Res. Commun., 506 (2018), 883–890. https://doi.org/10.1016/j.bbrc.2018.10.161 doi: 10.1016/j.bbrc.2018.10.161
    [37] K. M. McCarthy, I. B. Skare, M. C. Stankewich, M. Furuse, S. Tsukita, R. A. Rogers, et al., Occludin is a functional component of the tight junction, J. Cell Sci., 109 (1996), 2287–2298. https://doi.org/10.1242/jcs.109.9.2287 doi: 10.1242/jcs.109.9.2287
    [38] J. Sakata, T. Shimokubo, K. Kitamura, S. Nakamura, K. Kangawa, H. Matsuo, et al., Molecular cloning and biological activities of rat adrenomedullin, a hypotensive peptide, Biochem. Biophys. Res. Commun., 195 (1993), 921–927. https://doi.org/10.1006/bbrc.1993.2132 doi: 10.1006/bbrc.1993.2132
    [39] K. Miyashita, H. Itoh, N. Sawada, Y. Fukunaga, M. Sone, K. Yamahara, et al., Adrenomedullin promotes proliferation and migration of cultured endothelial cells, Hypertens. Res., 26 Suppl (2003), S93–98. https://doi.org/10.1291/hypres.26.S93 doi: 10.1291/hypres.26.S93
    [40] N. Ferrara, H. P. Gerber, J. LeCouter, The biology of VEGF and its receptors, Nat. Med., 9 (2003), 669–676. https://doi.org/10.1038/nm0603-669 doi: 10.1038/nm0603-669
    [41] M. I. Lin, W. C. Sessa, Vascular endothelial growth factor signaling to endothelial nitric oxide synthase: more than a FLeeTing moment, Circ. Res., 99 (2006), 666–668. https://doi.org/10.1161/01.RES.0000245430.24075.a4 doi: 10.1161/01.RES.0000245430.24075.a4
    [42] S. Sugo, N. Minamino, K. Kangawa, K. Miyamoto, K. Kitamura, J. Sakata, et al., Endothelial cells actively synthesize and secrete adrenomedullin, Biochem. Biophys. Res. Commun., 201 (1994), 1160–1166. https://doi.org/10.1006/bbrc.1994.1827 doi: 10.1006/bbrc.1994.1827
    [43] M. J. Miller, A. Martinez, E. J. Unsworth, C. J. Thiele, T. W. Moody, T. Elsasser, et al., Adrenomedullin expression in human tumor cell lines. Its potential role as an autocrine growth factor, J. Biol. Chem., 271 (1996), 23345–23351. https://doi.org/10.1074/jbc.271.38.23345 doi: 10.1074/jbc.271.38.23345
    [44] K. Dawas, M. Loizidou, A. Shankar, H. Ali, I. Taylor, Angiogenesis in cancer: the role of endothelin-1, Ann. R. Coll. Surg. Engl., 81 (1999), 306–310.
    [45] N. Zhu, L. Gu, J. Jia, X. Wang, L. Wang, M. Yang, W. Yuan, Endothelin-1 triggers human peritoneal mesothelial cells' proliferation via ERK1/2-Ets-1 signaling pathway and contributes to endothelial cell angiogenesis, J. Cell. Biochem., 120 (2019), 3539–3546. https://doi.org/10.1002/jcb.27631 doi: 10.1002/jcb.27631
    [46] Y. Katanasaka, Y. Kodera, Y. Kitamura, T. Morimoto, T. Tamura, F. Koizumi, Epidermal growth factor receptor variant type III markedly accelerates angiogenesis and tumor growth via inducing c-myc mediated angiopoietin-like 4 expression in malignant glioma, Mol. Cancer, 12 (2013), 31. https://doi.org/10.1186/1476-4598-12-31 doi: 10.1186/1476-4598-12-31
    [47] R. Kolb, P. Kluz, Z. W. Tan, N. Borcherding, N. Bormann, A. Vishwakarma, et al., Obesity-associated inflammation promotes angiogenesis and breast cancer via angiopoietin-like 4, Oncogene, 38 (2019), 2351–2363. https://doi.org/10.1038/s41388-018-0592-6 doi: 10.1038/s41388-018-0592-6
    [48] M. S. Airaksinen, M. Saarma, The GDNF family: signalling, biological functions and therapeutic value, Nat. Rev. Neurosci., 3 (2002), 383–394. https://doi.org/10.1038/nrn812 doi: 10.1038/nrn812
    [49] J. Kang, P. X. Qian, V. Pandey, J. K. Perry, L. D. Miller, E. T. Liu, et al., Artemin is estrogen regulated and mediates antiestrogen resistance in mammary carcinoma, Oncogene, 29 (2010), 3228–3240. https://doi.org/10.1038/onc.2010.71 doi: 10.1038/onc.2010.71
    [50] J. Kang, J. K. Perry, V. Pandey, G. C. Fielder, B. Mei, P. X. Qian, et al., Artemin is oncogenic for human mammary carcinoma cells, Oncogene, 28 (2009), 2034–2045. https://doi.org/10.1038/onc.2009.66 doi: 10.1038/onc.2009.66
    [51] A. Banerjee, Z. S. Wu, P. Qian, J. Kang, V. Pandey, D. X. Liu, et al., ARTEMIN synergizes with TWIST1 to promote metastasis and poor survival outcome in patients with ER negative mammary carcinoma, Breast Cancer Res., 13 (2011), R112. https://doi.org/10.1186/bcr3054 doi: 10.1186/bcr3054
    [52] A. Banerjee, P. Qian, Z. S. Wu, X. Ren, M. Steiner, N. M. Bougen, et al., Artemin stimulates radio- and chemo-resistance by promoting TWIST1-BCL-2-dependent cancer stem cell-like behavior in mammary carcinoma cells, J. Biol. Chem., 287 (2012), 42502–42515. https://doi.org/10.1074/jbc.M112.365163 doi: 10.1074/jbc.M112.365163
    [53] H. Zhang, C. C. Wong, H. Wei, D. M. Gilkes, P. Korangath, P. Chaturvedi, et al., HIF-1-dependent expression of angiopoietin-like 4 and L1CAM mediates vascular metastasis of hypoxic breast cancer cells to the lungs, Oncogene, 31 (2012), 1757–1770. https://doi.org/10.1038/onc.2011.365 doi: 10.1038/onc.2011.365
    [54] A. Banerjee, Z. S. Wu, P. X. Qian, J. Kang, D. X. Liu, T. Zhu, et al., ARTEMIN promotes de novo angiogenesis in ER negative mammary carcinoma through activation of TWIST1-VEGF-A signaling, PLoS One, 7 (2012), e50098. https://doi.org/10.1371/journal.pone.0050098 doi: 10.1371/journal.pone.0050098
    [55] A. Friedli, E. Fischer, Novak-Hofer I, S. Cohrs, K. Ballmer-Hofer, P. A. Schubiger, et al., The soluble form of the cancer-associated L1 cell adhesion molecule is a pro-angiogenic factor, Int. J. Biochem. Cell Biol., 41 (2009), 1572–1580. https://doi.org/10.1016/j.biocel.2009.01.006 doi: 10.1016/j.biocel.2009.01.006
    [56] H. Hall, J. A. Hubbell, Matrix-bound sixth Ig-like domain of cell adhesion molecule L1 acts as an angiogenic factor by ligating alphavbeta3-integrin and activating VEGF-R2, Microvasc. Res., 68 (2004), 169–178. https://doi.org/10.1016/j.mvr.2004.07.001 doi: 10.1016/j.mvr.2004.07.001
    [57] H. Hall, V. Djonov, M. Ehrbar, M. Hoechli, J. A. Hubbell, Heterophilic interactions between cell adhesion molecule L1 and alphavbeta3-integrin induce HUVEC process extension in vitro and angiogenesis in vivo, Angiogenesis, 7 (2004), 213–223. https://doi.org/10.1007/s10456-004-1328-5 doi: 10.1007/s10456-004-1328-5
    [58] M. Zhang, W. Zhang, Z. Wu, S. Liu, L. Sun, Y. Zhong, et al., Artemin is hypoxia responsive and promotes oncogenicity and increased tumor initiating capacity in hepatocellular carcinoma, Oncotarget, 7 (2016), 3267–3282. https://doi.org/10.18632/oncotarget.6572 doi: 10.18632/oncotarget.6572
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