Review Topical Sections

A survey on key roles of optical switching and labeling technologies on big data traffic of Data Centers and HPC environments

  • Received: 19 April 2019 Accepted: 10 June 2019 Published: 20 June 2019
  • The tremendous big data and IP traffic growth rate between interconnected Data Centers (DC) and High Performance Computing (HPC) environments have imposed the need for ultrahigh link capacities and ultrahigh packet switching speeds, at network nodes. In order to overcome these ultrahigh demands, and particularly packet routing and forwarding speeds, long tested and established technologies, such as optical switching and labeling technology, seem to provide adequate solutions, not only by conveying ultrahigh bit rate data streams, but also by achieving multi Tb/s cross connection throughputs, in a cost and energy efficient way. By adoption of optical switching and labeling technology, big data streams are propagating directly in optical layer, thus lessening down bottlenecks, latency issues, and multi stage hierarchy layering. This paper, apart from optical switching and labeling potentials, investigates thoroughly other critical issues, strictly related to the proper choice of employing a switching architecture layout, such as its implementation technology, its elasticity potentials, in terms of flexible bandwidth (BW) provisioning, its adopted control plane lying on top of data infrastructure plane, providing cognition, control and orchestration over its network elements, as well as related to the proper choice of optical labeling techniques adopted, in conjunction with current, advanced, coherent, multi level modulation formats, for ultrahigh link capacities and packet switching speed demands of scalable, big data interconnected DCs and HPC environments.

    Citation: Efthymios N. Lallas. A survey on key roles of optical switching and labeling technologies on big data traffic of Data Centers and HPC environments[J]. AIMS Electronics and Electrical Engineering, 2019, 3(3): 233-256. doi: 10.3934/ElectrEng.2019.3.233

    Related Papers:

    [1] Ryo Kinoshita, Sung-mok Jung, Tetsuro Kobayashi, Andrei R. Akhmetzhanov, Hiroshi Nishiura . Epidemiology of coronavirus disease 2019 (COVID-19) in Japan during the first and second waves. Mathematical Biosciences and Engineering, 2022, 19(6): 6088-6101. doi: 10.3934/mbe.2022284
    [2] Avinash Shankaranarayanan, Hsiu-Chuan Wei . Mathematical modeling of SARS-nCoV-2 virus in Tamil Nadu, South India. Mathematical Biosciences and Engineering, 2022, 19(11): 11324-11344. doi: 10.3934/mbe.2022527
    [3] Yukun Tan, Durward Cator III, Martial Ndeffo-Mbah, Ulisses Braga-Neto . A stochastic metapopulation state-space approach to modeling and estimating COVID-19 spread. Mathematical Biosciences and Engineering, 2021, 18(6): 7685-7710. doi: 10.3934/mbe.2021381
    [4] Christian Costris-Vas, Elissa J. Schwartz, Robert Smith? . Predicting COVID-19 using past pandemics as a guide: how reliable were mathematical models then, and how reliable will they be now?. Mathematical Biosciences and Engineering, 2020, 17(6): 7502-7518. doi: 10.3934/mbe.2020383
    [5] Qian Shen . Research of mortality risk prediction based on hospital admission data for COVID-19 patients. Mathematical Biosciences and Engineering, 2023, 20(3): 5333-5351. doi: 10.3934/mbe.2023247
    [6] Antonios Armaou, Bryce Katch, Lucia Russo, Constantinos Siettos . Designing social distancing policies for the COVID-19 pandemic: A probabilistic model predictive control approach. Mathematical Biosciences and Engineering, 2022, 19(9): 8804-8832. doi: 10.3934/mbe.2022409
    [7] Damilola Olabode, Jordan Culp, Allison Fisher, Angela Tower, Dylan Hull-Nye, Xueying Wang . Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China. Mathematical Biosciences and Engineering, 2021, 18(1): 950-967. doi: 10.3934/mbe.2021050
    [8] Enahoro A. Iboi, Oluwaseun Sharomi, Calistus N. Ngonghala, Abba B. Gumel . Mathematical modeling and analysis of COVID-19 pandemic in Nigeria. Mathematical Biosciences and Engineering, 2020, 17(6): 7192-7220. doi: 10.3934/mbe.2020369
    [9] Xinyu Bai, Shaojuan Ma . Stochastic dynamical behavior of COVID-19 model based on secondary vaccination. Mathematical Biosciences and Engineering, 2023, 20(2): 2980-2997. doi: 10.3934/mbe.2023141
    [10] Hamdy M. Youssef, Najat A. Alghamdi, Magdy A. Ezzat, Alaa A. El-Bary, Ahmed M. Shawky . A new dynamical modeling SEIR with global analysis applied to the real data of spreading COVID-19 in Saudi Arabia. Mathematical Biosciences and Engineering, 2020, 17(6): 7018-7044. doi: 10.3934/mbe.2020362
  • The tremendous big data and IP traffic growth rate between interconnected Data Centers (DC) and High Performance Computing (HPC) environments have imposed the need for ultrahigh link capacities and ultrahigh packet switching speeds, at network nodes. In order to overcome these ultrahigh demands, and particularly packet routing and forwarding speeds, long tested and established technologies, such as optical switching and labeling technology, seem to provide adequate solutions, not only by conveying ultrahigh bit rate data streams, but also by achieving multi Tb/s cross connection throughputs, in a cost and energy efficient way. By adoption of optical switching and labeling technology, big data streams are propagating directly in optical layer, thus lessening down bottlenecks, latency issues, and multi stage hierarchy layering. This paper, apart from optical switching and labeling potentials, investigates thoroughly other critical issues, strictly related to the proper choice of employing a switching architecture layout, such as its implementation technology, its elasticity potentials, in terms of flexible bandwidth (BW) provisioning, its adopted control plane lying on top of data infrastructure plane, providing cognition, control and orchestration over its network elements, as well as related to the proper choice of optical labeling techniques adopted, in conjunction with current, advanced, coherent, multi level modulation formats, for ultrahigh link capacities and packet switching speed demands of scalable, big data interconnected DCs and HPC environments.


    The novel coronavirus SARS-CoV-2 has caused a global pandemic of unprecedented viral pneumonia [1,2]. This infection is known as coronavirus disease 2019 (COVID-19) [3]. Because of the high human-to-human transmissibility, SARS-CoV-2 has spread rapidly around the world [4,5,6,7]. In mainland China, the outbreak started in December 2019, reached the peak in February and then the number of new confirmed cases decreased. On March 18, 2020, there were no new cases of infection for the first time and the economy and daily life gradually returned to normal [8,9,10,11]. In Hong Kong Special Administrative Region (SAR), the first case was reported on January 18, 2020. The outbreak peaked in late March 2020 with no new confirmed cases on April 23, 2020. However, spread of COVID-19 in the world is continuing and the outbreak is ongoing globally [12,13]. The infection has been confirmed in about 190 countries up to now. As of February 21, 2021, there have been over 110.6 million cases and 2.45 million deaths reported globally since the beginning of the pandemic. The European Region has the largest new cases and new deaths. The United States accounts for the greatest proportion of cumulative cases and deaths [14]. In China, although the epidemic has been under control, confirmed cases have been found occasionally in different places, which raised significant concerns on the resurgence of future waves of COVID-19.

    If there are no confirmed cases in a region for a long time, then the risk of disease re-emergence might be mainly from imported cases or viruses. The major COVID-19 transmission pathway is from human to human through respiratory droplets [15,16]. In particular, asymptomatic individuals who do not have COVID-19 symptoms can still spread the virus. Transmission from asymptomatic individuals poses a significant public health challenge [17,18,19,20,21,22,23]. The cases imported from other high-risk places are another path of viral spread. To reduce the potential of imported cases, many countries have issued travel restrictions, for example, reducing the frequency of flights from abroad [24,25]. However, as the infection is still prevailing in many places, imported cases still represent a tremendous risk, which may lead to new local outbreaks [26,27,28,29]. Another possible path of SARS-CoV-2 transmission might be through the food supply chain, surfaces and environment. In China, the coronavirus was detected on frozen foods, including their packaging materials and storage environment in July 2020. There seemed to have two outbreaks related to the transmission via frozen food [30]. In view of this, interventions that reduce foodborne transmission of pathogens need to be considered [32].

    Non-pharmaceutical control measures implemented so far are mainly wearing mask, hand washing, social distancing, quarantine and city/region lockdown [31]. These interventions were gradually lifted in consideration of the trade-off between economic sustainability and public health. An agent-based model was developed to evaluate the possibility of a second-wave emergence under different extents and timing of intervention relaxation [32]. More work assessed the risk of secondary waves since the control measures like lockdowns were relaxed [33,34,35,36,37,38,39,40]. The study [41] compared the data of the epidemiological pattern of COVID-19 in 53 countries or regions where the pandemic experienced two waves, and analyzed the differences between the two outbreaks. Their results suggested that there was a shift of infection to younger age groups, which may make it more difficult to control the pandemic.

    In this work, we focus on the COVID-19 spread in serval places where the epidemic has been under control but new cases have been reported occasionally. To study the impact of imported cases on the dynamics of COVID-19 in China under different scenarios of prevention and control measures, Jia et al. developed an impulsive epidemic model to describe imported cases from abroad [42]. In their model, the time when the exposed cases were imported was fixed. However, the exposed cases who carry virus without symptoms are usually unknown. When an infected case is identified, the virus has probably been spreading for a period of time. In the beginning of a new wave of epidemic, the infection might be induced by a small number of infected cases. The disease transmission in this stage can be affected by many random factors. In addition, the data of new/accumulated cases were reported on every day. All of these motivate us to develop a stochastic discrete-time compartmental model that considers randomness, epidemic data, as well as the impact of input virus/cases and the initial entry time. By fitting the model to the two waves of outbreaks in two places in mainland China (Beijing and Xinjiang) and Hong Kong SAR, we evaluate the risk factors that can affect the second or future wave of COVID-19.

    We develop a stochastic discrete-time model based on the classic compartmental model. Individuals who have no clinical manifestations such as fever, cough, sore throat and other symptoms that can be self-perceived or clinically recognized, but test positive in serological or blood test are referred to as asymptomatic infection. This population includes two types of individuals. One is asymptomatic infection in the incubation period. They will later develop clinical symptoms or become a confirmed case by screening test or CT (Computed Tomography) examination. The other has no symptoms until the nucleic acid test turns negative. The total population is divided into five epidemiological classes, including susceptible (S), exposed (E), asymptomatically infected (A), symptomatically infected (I), and recovered (R). Due to quarantine, the susceptible and exposed states are further divided into Sq and Eq. With hospitalization, the infected class (both asymptomatic and symptomatic) can be further divided into HA and HI. Because the infection and disease progression can be affected by random factors, we assume that the flow between any two compartments is a stochastic process [43,44,45]. For example, D11(t) is the number of susceptible individuals who become newly infected and this process obeys a binomial distribution. The diagram of the model is shown in Figure 1 and the corresponding stochastic discrete-time model is given by the following system:

    St+1=StD11(t)D12(t)+D51(t),Et+1=Et+(1q)D11(t)D21(t),At+1=At+(1ρ)D21(t)D31(t)D32(t)D33(t),It+1=It+ρD21(t)D41(t)D42(t)D43(t)+D33(t),Sqt+1=Sqt+D12(t)D51(t),Eqt+1=Eqt+qD11(t)D61(t),HAt+1=HAt+D31(t)+(1ρ)D61(t)D71(t)D72(t),HIt+1=HIt+D41(t)+ρD61(t)+D71(t)D81(t)D82(t),Rt+1=Rt+D32(t)+D42(t)+D72(t)+D82(t), (2.1)
    Figure 1.  Flow diagram of the COVID-19 infection model.

    where Dij(t) obeys a binomial distribution Bin(n,p) with the parameters (n,p), and the specific form is as follows

    D11(t)Bin(St,P11(t)),  D12(t)Bin(St,P12(t)),  D21(t)Bin(Et,P21),D31(t)Bin(At,P31(t)),  D32(t)Bin(At,P32),  D33(t)Bin(At,P33),D41(t)Bin(It,P41),  D42(t)Bin(It,P42),  D43(t)Bin(It,P43),D51(t)Bin(Sqt,P51),  D61(t)Bin(Eqt,P61),  D71(t)Bin(HAt,P33),D72(t)Bin(HAt,P32),  D81(t)Bin(HIt,P43)  D82(t)Bin(HIt,P42).
    P11(t)=1exp[βc(t)(I+θA)N],  P12(t)=1exp[(1β)qc(t)(I+θA)N],P21=1exp(σ),  P31=1exp(δA),P32(t)=1exp(γA),  P33=1exp(k),P41=1exp(δI),  P42=1exp(γI),P51=1exp(λ),  P61=1exp(δq).

    Here exp[βc(t)(I+θA)Nh] is the probability of staying in the compartment S. The time period h is chosen to be one, so it is omitted in the expression. Thus, P11 is the probability of individuals leaving the susceptible compartment. The other P functions can be explained in a similar way. The meaning of each parameter in the model is summarized in Table 1.

    Table 1.  Parameter estimates for the COVID-19 epidemics in Beijing, Xinjiang and Hong Kong SAR.
    Parameters Definition Values Source
    Beijing Xinjiang Hong Kong SAR
    c(t) c0 Contact rate at the initial time 17.0142 14.0025 11.9175 Estimated
    cb Minimum contact rate under the most strict control strategies 1.0869 2.0468 3.0460 Estimated
    r1 Exponential decreasing rate of contact rate in the first period 0.2196 0.2049 0.0703 Estimated
    r2 Exponential increasing rate of contact rate in the second period 0.0506 0.0200 0.0324 Estimated
    r3 Exponential decreasing rate of contact rate in the third period 0.2241 0.3096 0.1445 Estimated
    β Probability of transmission per contact 0.2801 0.2722 0.1504 Estimated
    q(t) q0 Quarantined rate at the initial time 0.2820 Estimated
    qm Maximum quarantined rate with control strategies 0.7083 Estimated
    r4 Exponential increasing rate of quarantined rate in the first period 0.2027 Estimated
    r5 Exponential decreasing rate of quarantined rate in the second period 0.1010 Estimated
    r6 Exponential increasing rate of quarantined rate in the third period 0.2028 Estimated
    q Quarantined rate 0.2935 0.4866 Estimated
    ρ Ratio of symptomatic infection 0.5142 0.5520 0.5529 Estimated
    σ Transition rate of exposed individuals to the infected class 1/5 1/5 1/5 [47]
    λ Rate at which the quarantined uninfected contacts were released into the wider community 1/14 1/14 1/14 [47]
    δI Constant transition rate of symptomatic infected 0.3474 0.2008 0.0999 Estimated
    δA Constant transition rate of asymptomatic infected 0.2860 0.2006 0.3027 Estimated
    δq Constant transition rate of quarantined exposed 0.3599 0.2816 0.2571 Estimated
    θ Correction factor for transmission probability of asymptomatic infectious 0.5919 0.5031 0.5041 Estimated
    k Conversion rate from asymptomatic infected to symptomatic infected 0.6124 0.6221 0.5026 Estimated
    γI Recovery rate of infected individuals 0.0701 0.1632 0.0799 Estimated
    γA Recovery rate of asymptotic infected individuals 0.0906 0.1629 0.2393 Estimated
    τ The time of importation of the first case in the second wave 17 16 Estimated
    pE(T2τ) The number of exposed cases entered at the time T2τ in the second wave 6 7.02 Estimated
    α Disease-induced death rate 0 0 0 Assumed
    – means the parameter is not included in that place.

     | Show Table
    DownLoad: CSV

    Due to limited pharmaceutical interventions, wearing mask and social distancing play a critical role in the control of the COVID-19 pandemic. As the epidemic is gradually controlled, people's vigilance will decrease. Strict intervention measures may have to be lifted because of economic consideration. We use a time-varying function for the contact rate to describe this change. When the pandemic began and spread rapidly, control measures such as city lockdown, wearing masks and social activity reduction greatly reduced the contact between people. We denote the time of strict control implementation by T0. When the number of infected cases gradually decrease after the peak, the control measures are relaxed and people's lives gradually return to normal. We denote this time by T1. When new cases are reported again, people's vigilance increases, and prevention and control measures are implemented again. We denote this time by T2. The following time-varying function for the contact rate c(t) is used to describe the change of human behavior and effect of control measures during the epidemic.

    c(t)={c0,t<T0,(c0cb)er1(tT0)+cb,T0t<T1,(cb1c0)er2(tT1)+c0,T1t<T2,(c01cb)er3(tT2)+cb,tT2. (2.2)

    Here cb1=(c0cb)er1(T1T0)+cb and c01=(cbc0)er2(T2T1)+c0.

    We define the quarantine rate q(t) in a similar way. The quarantine rate increases as the epidemic gets worse and decreases as it improves. Thus, we assume that the quarantine rate is a time-dependent piecewise function, given by

    q(t)={q0,t<T0,(q0qm)er4(tT0)+qm,T0t<T1,(qm1q0)er5(tT1)+q0,T1t<T2,(q01qm)er6(tT2)+qm,tT2, (2.3)

    where qm1=(q0qm)er4(T1T0)+qm and q01=(qmq0)er5(T2T1)+q0. The functions c(t) and q(t) are shown in Figure 2(a–b), Figure 3(e–f) and Figure 4(a–b) for three different places.

    Figure 2.  The fitting of model (2.1) to the data of COVID-19 in Beijing from January 26 to July 31 in 2020. (a-c): the change of contact rate c(t), quarantine rate q(t), and the total population N(t), respectively. The daily confirmed cases are shown in (d), daily asymptomatic cases are shown in (e), recovered cases are shown in (f). Stochastic fit was performed 50 times with light blue lines. The data and the fitting are represented by black points and the deep blue lines, respectively.
    Figure 3.  The fitting of model (2.1) to the data of COVID-19 in Xinjiang from January 23 to August 18, 2020. The daily confirmed cases are shown in (a), daily asymptomatic new cases are shown in (b), daily asymptomatic cases are shown in (c), cumulative confirmed cases are shown in (d). Stochastic fit was performed 50 times with light blue lines. The data and the fitting are represented by black points and deep blue lines, respectively. (e-g): the change of contact rate c(t), quarantine rate q(t), and the total population N(t), respectively.
    Figure 4.  The fitting of model (2.1) to the data of COVID-19 in Hong Kong SAR from February 23 to July 31, 2020. (a-c): the change of contact rate c(t), quarantine rate q(t), and the total population N(t), respectively. The daily confirmed cases are shown in (d), recovered cases are shown in (e). Stochastic fit was performed 50 times with light blue lines. The data and the fitting are represented by black points and deep blue lines, respectively.

    We collected the data of Beijing and Xinjiang from the local health commissions in mainland China, and the data of Hong Kong SAR from the Centre for Health Protection. It includes the time series data of confirmed COVID-19 cases, recovered cases, and asymptomatic coronavirus carriers. On December 26, 2019, a respiratory and critical care physician in Wuhan reported the pneumonia of unknown cause for the first time. The epidemic then spread rapidly in mainland China, and the number of newly confirmed cases reached the peak on February 4, 2020. As of March 18, the number of newly confirmed cases in mainland China became 0 and the number of confirmed cases fell below 20,000. After that, the reported cases in mainland China were mainly imported cases. A few months later, infected cases began to rise again in some places. On June 11, 2020, a confirmed case was reported in Beijing, without history of traveling outside Beijing and without close contact with suspected infection in the past two weeks. This ended a 56-day streak of no local infection in Beijing. On July 15, 2020, i.e., 149 days since the previous confirmed cases, one confirmed case and three asymptomatic cases were found in Xinjiang. In Hong Kong SAR, there were sporadic confirmed cases after April 20. On July 5, a second-wave outbreak emerged. This paper will focus on the data from these three places to study the risk of the emergence of a future wave of COVID-19. The switching time T0, T1 and T2 in the piecewise function are determined by the responding time in each place.

    If there are no cases for a long period of time, e.g., several months, after a wave of COVID-19 outbreak, then the new infection is likely to be caused by imported cases or exposure to the virus. The virus that caused a second wave can be summarized by the following three sources: (1) imported cases from abroad. Despite strict regulations on international travel and border inspections, there are still some reported cases imported from abroad. There is no guarantee that 100% of the infected or exposed cases entering the country will be isolated. The incubation period of the infection is not well known and may not be the same for all infected people. With fixed-duration quarantine implemented, the infected individual may become a confirmed case after the quarantine is over. This may be a risk for a second wave in mainland China. (2) Asymptomatic cases. These people carry the virus but cannot be identified if they do not have the nucleic acid test. However, they can infect other people. Therefore, asymptomatic carriers represent another risk for the occurrence of the second wave. (3) Virus from the environment. Some studies have shown that low temperature can greatly promote the persistence of coronaviruses. Frozen foods are potential carriers. Transmission occurs via touching contaminated objects that mediate the infection through mouth, nose, or eyes. This seems to be another risk of transmission that have been ignored.

    The potential causes summarized above can be described by new exposed individuals added to our model at a certain time. The time point when the new confirmed case was reported is T2 but when the exposed individual was introduced remains unknown. Here we assume that the number of input exposed individuals is pE(T2τ) where τ represents the time lag from the entry of the exposed individual to the later confirmation of infection. Thus, T2τ is the time point when the exposed individuals entered. The equation of E(t) in model (2.1) can be replaced by the following equation

    Et+1=Et+(1q)D11(t)D21(t)+pE(T2τ).

    It is noted that the reported case and the imported case may not be the same person.

    The increase in the susceptible population due to lifted interventions may also contribute to the second wave. The first wave of COVID-19 emerged in Wuhan in early January of 2020. The time happened to be about ten days before the Lunar New Year. This made most people stay at home and take the longest vacation, which greatly reduced the probability of contact. In addition, public transportation was terminated and schools and restaurants were all closed. This series of strict measures reduced the number of susceptible people to a very small level. In our model, we assume that the number of susceptible people in the first wave of outbreak is S01. After the first wave, social activities gradually returned to normal and the size of susceptible population increases to S02 when the second wave emerges. The time of the susceptible population change, denoted by T3, depends on the region. For Beijing and Xinjiang, we let it be the same as T1. For Hong Kong SAR, it is the time when the local restriction policy is released. Thus, the number of susceptible is given by the following piecewise function

    S0={S01,t<T3,S02,tT3. (2.4)

    We use the discrete stochastic model (2.1) with the input parameter pE(T2τ) to fit the data of the two waves of outbreaks in Beijing and Xinjiang using the least square method. The data fitted include the number of reported confirmed cases, asymptomatic cases and recovered cases. For the epidemic in Hong Kong SAR, there were still sporadic reports of confirmed cases after the first wave. The reason for the second wave in Hong Kong SAR is likely the increase in the number of susceptible population due to lifted restriction of interventions. We use the model (2.1) without the input parameter pE(T2τ) to fit the data in Hong Kong SAR. Parameter values obtained from the fitting are listed in Tables 1 and 2. The population size of susceptible in the three places is less than the entire population of those places. Here the susceptible population refers to those who may contact with the infected cases. The stochastic simulations provide good fits to the data in these three palaces, see Figure 2(d–f), Figure 3(d–f) and Figure 4(d–f). The corresponding contact rate c(t), quarantine rate q(t) and the susceptible population change S0(t) are shown in Figure 2(a–c), Figure 3(e–g) and Figure 4(a–c), respectively.

    Table 2.  Initial values for the COVID-19 epidemics in Beijing, Xinjiang and Hong Kong SAR.
    Initial values Definition Values Source
    Beijing Xinjiang Hong Kong SAR
    S01 The value of susceptible population in the first wave 5.0014×103 9.4767×103 7.3327×103 Estimated
    S02 The value of the susceptible in the second wave 6.0119×103 4.9377×104 2×104 Estimated
    E(0) The initial value of exposed population 8.0747 8.0209 6.0401 Estimated
    I(0) The initial value of infected symptomatic population 4.0902 3.0368 6.0316 Estimated
    A(0) The initial value of infected asymptomatic population 5.0848 4.0312 2.0328 Estimated
    Sq(0) The initial value of quarantined susceptible population 49.7473 49.9653 45.7412 Estimated
    Eq(0) The initial value of quarantined exposed population 20.0155 5.0235 13.1394 Estimated
    HI(0) The initial value of confirmed and hospitalized symptomatic population 1 3 5 Data
    1 3 5 Data
    HA(0) The initial value of confirmed and hospitalized asymptomatic population 0 1 0 Data
    0 1 0 Data
    R(0) The initial value of recovered population 0 0 0 Data

     | Show Table
    DownLoad: CSV

    The emergence of the second wave is influenced by the number of input exposed individuals and how long the infection has been spreading before the report of confirmed cases. We conduct numerical simulations to study the risk of having a second wave. The occurrence of a second wave is evaluated by the maximum number of confirmed cases in 500 stochastic simulations. We denote the average number by MH and choose a threshold value 30. If the MH value exceeds 30, it will be regarded as a second wave. The result shows that not all the scenarios result in a second wave. From 500 stochastic simulations, we calculate the probability of the occurrence of a second wave, which is denoted by Prop.

    In Figures 5 and 6, we explore the effect of varying the input parameter on the risk of second wave in Beijing and Xinjiang, respectively. The range of the parameter pE is set to [0,30] at time T2τ, and the time delay parameter τ is within the range [0,20]. From Figure 5(a), we find that both the number of input exposed individuals and the time between initial entry and subsequent confirmation affect the severity of the second wave. The average maximum value of the second wave peak can reach 1600 cases in Beijing. Increasing the number of input exposed individuals can expand the scale of the disease spread. A larger time delay τ implies that the disease had spread for a longer time without any interventions before its detection. This poses a substantial challenge for the subsequent control of the disease.

    Figure 5.  Maximum cases and the probability of occurrence of the second wave with different imported cases and time to detection in Beijing. (a, c): the results when imported cases are 1 to 30 from top to bottom. The time delay before detection is 1 to 20 from left to right. (b): The red point means that the maximum cases MH excess the threshold 30 and the blue point means MH is below the threshold.
    Figure 6.  Maximum cases and the probability of occurrence of the second wave with different imported cases and time to detection in Xinjiang. The other is the same as that in Figure 5.

    We provide the parameter region of a second wave occurrence in Figure 5(b). The deep blue points represent the parameter range of the occurrence of a second wave, while the deep red points represent the parameter range of no second wave. The simulation shows that a second outbreak would not take place when less than three exposed cases were imported. If the infection induced by the imported cases can be quickly identified, then the chance of having a second wave decreases. Figure 5(c) further shows the probability of the occurrence of the second wave under the same parameter range in Figure 5(a). Large values of pE and τ will make a second wave inevitable. We have the similar conclusion from the simulation for Xinjiang (see Figure 6). The scale of the second wave is larger than Beijing with the same parameter range because the average maximum value of possible second wave peak can reach 5000 cases in the worst scenario.

    Figure 9(a) shows the average result of 500 stochastic simulations of the model (2.1) with six different susceptible populations in Hong Kong SAR. As the susceptible population increases, the average maximum value of the second wave peak also increases. Interestingly, the probability of the occurrence of the second wave remains almost the same for different susceptible populations (see Figure 9(b)). Numerical results on the effect of varying the susceptible population size in Beijing and Xinjiang are shown in Figures 7 and 8, respectively. Based on the simulations in these two places, we have a conclusion similar to Hong Kong SAR. This analysis suggests that the susceptible population size plays a minor role in leading to the second wave when the other parameters are fixed.

    Figure 7.  (a) Simulation of confirmed cases under six scenarios in Beijing when the susceptible population is chosen to be 0.5S01, 0.8S01, S01, S02, 1.2S02 and 1.5S02. (b) The probability of occurrence of a second wave under the six scenarios.
    Figure 8.  Simulation of confirmed cases under five scenarios in Xinjiang when the susceptible population is chosen to be 0.5S01, S01, middle value of S01 and S02, S02, 1.5S02. (b) The probability of occurrence of a second wave under the five scenarios.
    Figure 9.  Simulation of confirmed cases under five scenarios in Hong Kong SAR when the susceptible population is chosen to be 0.5S01, S01, middle value of S01 and S02, S02, 1.5S02. (b) The probability of occurrence of a second wave under the five scenarios.

    COVID-19, a highly contagious disease first reported in December 2019, has been spreading globally for more than one year. Some countries/regions have mitigated the outbreak by various measures but are still at risk of recurrence. In this paper, we constructed a stochastic discrete-time compartmental epidemic model to analyze the risk of the occurrence of a second or future wave of outbreak. Compared with a deterministic system, a stochastic model is able to include the random factors in the spread of an infectious disease, particularly when the number of initial infected individuals is small. This is the case when a new wave of outbreak occurs. This discrete model can more intuitively describe the flow between any two compartments. The transition between two compartments is not deterministic and assumed to obey binomial distributions in our model. The change between two compartments in one time step corresponds to the daily data. Thus, using the discrete stochastic model facilitates full use of the data from multiple sources, thereby improving the reliability of the parameter estimation results.

    To describe the change in the intensity of control measures in response to the COVID-19 pandemic, we adopt time-varying contact rate and quarantine rate in the model. There are a few possible factors that may lead to a second wave, including import exposed cases, asymptomatic cases, and the presence of viruses in the environment such as the frozen food chain. The common characteristic of these factors is that the transmission is silent and difficult to be identified. We find that the time between the exposed case entry and the confirmation of subsequent infection plays a critical role in the occurrence of the second wave.

    The cause of the second-wave outbreak in Beijing and Xinjiang is mainly the imported cases and an increase in the susceptible population due to relaxed interventions. The model provided a good fit to the data of the second wave in Beijing in June 2020. Based on the fitting, the value of input exposed cases is estimated to be 6 and the time from exposed individual entry to the detection of infection is 17 days. The size of susceptible population increases from 5.001×103 in the first wave to 6.012×103 in the second wave. For Xinjiang where the second wave of the epidemic occurred in July 2020, the value of input exposed cases is estimated to be 7 and the time from entry to detection is 16 days. The change in the number of susceptible people is greater than in Beijing.

    Hong Kong SAR also experienced a second wave in July 2020. Unlike Beijing and Xinjiang, there were occasional reports of infected cases all the time in Hong Kong SAR after the first wave and the main cause of the second wave is likely to be the increase in the number of susceptible people. Our modeling result suggests that in a region where the infection is not cleared (e.g., in Hong Kong SAR) susceptible people will increase as the control measures are lifted and this may lead to a second wave. If there is no case for a long time (e.g., in Beijing and Xinjiang), it is necessary to screen imported cases and viruses (e.g., via the food chain), which may be the major cause of the second wave.

    On the basis of the fitting to the data in Beijing and Xinjiang, we further evaluated the possibility of having a future outbreak and its severity. Because there were no confirmed cases for a long time after the first wave in Beijing and Xinjiang, the contact rate returned to the normal level, as shown in Figures 2 and 3. If there are imported exposed cases, the time to detect the infection is shown to be critical in leading to the second wave. The simulation shown in Figure 5 and Figure 6 indicates that the second wave is determined by the number of imported exposed individuals and the time needed to detect them. The results suggest that if the imported exposed cases are less than three, then the number of confirmed cases will be below the threshold 30 we set, which would not be considered as a second wave. If the values of imported exposed individuals and the time lag in detection are larger (e.g., in the red region in Figures 5 and 6), a second wave will emerge. The more imported exposed cases and the longer for the infection to be detected, the more likely a second wave will occur. Once a confirmed case is found, it is imperative to track the trajectory of that case and the contacted persons. The information obtained from this study can be used to evaluate the possibility (i.e., the possibility of infected cases above a threshold level) and scale of a future wave of outbreak.

    By investigating the effect of the susceptible population on the second wave in Beijing, Xinjiang and Hong Kong SAR, we found that the larger the susceptible population size, the more infections if the second wave occurs. However, the susceptible population size itself does not affect the probability of the occurrence of a second wave. This result suggests that imported cases might be an important factor leading to the occurrence of a second wave in a place where the epidemic has been well controlled. Once a case is found, reducing the number of susceptible people will help control the disease spread in the second wave.

    Our study cannot predict when a second or future wave of COVID-19 would take place. When a new wave occurs, the model can be used to predict the scale or severity of the outbreak. This is based on the fitting of the model to existing data. If the data are not sufficient for fitting, then the power of the model prediction would be limited. Lastly, the model does not include the influence of vaccination on the disease spread. How the vaccine rollout influences the emergence of future waves remains to be further investigated.

    In summary, we established a stochastic modeling framework that incorporates control measures at different stages of the epidemic and potential causes for the second wave emerged in Beijing, Xinjiang, and Hong Kong SAR. Because infected people without symptoms are contagious and the virus attached to goods is difficult to be detected, comprehensive measures are still imperative to curb the COVID-19 pandemic. It is necessary to screen the imported cases in flights and to detect the virus that may be transported by various routes. If a confirmed case is found, the contact of the case should be thoroughly tracked and the close contacts should be quarantined. Finally, it is important to continue protective measures such as wearing masks and avoiding large-scale gathering to reduce the number of susceptible people. This will make the future wave of outbreak less severe if it takes place.

    This work was finished when the first author visited the University of Florida in 2020. This research was partially supported by the National Natural Science Foundation of China (grant numbers: 12031010(ST), 11631012(ST)) and by the Fundamental Research Funds for the Central Universities (grant numbers: 2018CBLZ001(SH), GK201901008(ST)). L. Rong is supported by the National Science Foundation (grant number: DMS-1950254).

    No conflict of interest.



    [1] Khan N, Yaqoob I, Hashem IAT, et al. (2014) Big Data: Survey, Technologies, Opportunities, and Challenges. The Scientific World Journal 2014: 712826.
    [2] Vahdat A, Al-Fares M, Farrington N, et al. (2010) Scale-Out Networking in the Data Center. IEEE Micro 30: 29–41.
    [3] Miao WW, Yan FF and Calabretta NN (2016) Towards Petabit/s All-Optical Flat Data Center Networks Based on WDM Optical Cross-Connect Switches with Flow Control. IEEE Journal of Lightwave Technology 34: 4066–4075. doi: 10.1109/JLT.2016.2593040
    [4] Saridis GM, Aguado A, Yan Y, et al. (2018) LIGHTNESS: All-Optical SDN-enabled Intra-DCN with Optical Circuit and Packet Switching. Optical Switching in Next Generation Data Centers: 147–165.
    [5] Lallas EN, Xenakis A, Stamoulis G, et al. (2018) QoS and MPLS design issues in NoCs. In: 2018 South-Eastern European Design Automation, Computer Engineering, Computer Networks and Society Media Conference (SEEDA-CECNSM ).
    [6] Kitayama K, Huang YC, Yoshida Y, et al. (2014) Optical packet and path switching intra-data center network: Enabling technologies and network performance with intelligent flow control. The European Conference on Optical Communication, ECOC 2014, 21–25.
    [7] Gerstel O, Jinno M, Lord A, et al. (2012) Elastic optical networking: A new dawn for the optical layer?" IEEE Commun Mag 50: 12–20.
    [8] Lara A, Kolasani A and Ramamurthy B (2014) Network innovation using OpenFlow: A survey. IEEE Commun Surv Tut 16: 493–512. doi: 10.1109/SURV.2013.081313.00105
    [9] Kreutz D, Ramos FMV, Verissimo PJE, et al. (2015) Software-defined networking: A comprehensive survey. P IEEE 103: 14–76. doi: 10.1109/JPROC.2014.2371999
    [10] Qian D, Huang MF, lp E, et al. (2011) 101.7-Tb/s (370×294-Gb/s) PDM-128QAM-OFDM transmission over 3×55-km SSMF using pilot-based phase noise mitigation. In: 2011 Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference, 1–3.
    [11] Singh A, Ong J, Agarwal A, et al. (2016) Jupiter rising: a decade of Clos topologies and centralized control in Google's datacenter network. Communications of the ACM 59: 88–97.
    [12] Zhou X, Liu H, and Urata R (2017) Datacenter optics: requirements, technologies, and trends (Invited Paper). Chin Opt Lett 15: 120008–120011. doi: 10.3788/COL201715.120008
    [13] Cheng Q, Bahadori M, Glick M, et al. (2018) Recent advances in optical technologies for data centers: a review. Optica 5: 1354–1370. doi: 10.1364/OPTICA.5.001354
    [14] Kerravala Z (2015) A Data Center Fabric Is Critical to a Next-Generation Unified Data Center. Yankee Group White Paper.
    [15] Tucker RS (2011) Green optical communications-Part II: Energy limitations in networks. IEEE J Sel Top Quant 17: 261–274. doi: 10.1109/JSTQE.2010.2051217
    [16] Kitayama KI, Huang YC, Yoshida Y, et al. (2015) Torus-Topology Data Center Network Based on Optical Packet/Agile Circuit switching with Intelligent Flow Management. J Lightwave Technol 33: 1063–1071. doi: 10.1109/JLT.2015.2394384
    [17] Yin Y, Proietti R, Ye X, et al. (2013) LIONS: An AWGR-Based Low-Latency Optical Switch for High-Performance Computing and Data Centers. IEEE J Sel Top Quant 19: 3600409–3600409. doi: 10.1109/JSTQE.2012.2209174
    [18] Imran M, Collier M, Landais P, et al. (2015) HOSA: Hybrid Optical Switch Architecture for Data Center Networks. In: Proceedings of the 12th ACM International Conference on Computing Frontiers, p. 27.
    [19] Peng S, Simeonidou D, Zervas G, et al. (2014) A novel SDN enabled hybrid optical packet/circuit switched data centre network: The LIGHTNESS approach. In: 2014 European Conference on Networks and Communications (EuCNC), pp. 1–5.
    [20] Xu M, Diakonikolas J, Modiano E, et al. (2019) A Hierarchical WDM-based Scalable Data Center Network Architecture. ArXiv:1901.06450.
    [21] Kitayama K (2016) Optical Packet Switching: Myth, Fact, and Promise. In: 21st OptoElectronics and Communications Conference (OECC) held jointly with 2016 International Conference on Photonics in Switching (PS), pp. 1–2.
    [22] Ye S, Shen Y, and Panwar SS(2010) HELIOS: A High Energy-efficiency Locally-scheduled Input-queued Optical Switch. In: Proceedings of the 6th ACM/IEEE Symposium on Architectures for Networking and Communications Systems (ANCS), IEEE.
    [23] Balanici M and Pachnicke S (2018) Intra-Data Center Network Optimization by Means of Application-Based Traffic Modeling and Optical Circuit Switching. In: 20th International Conference on Transparent Optical Networks (ICTON), Bucharest, Romania.
    [24] Lone S, Usman M, Imran M (2018) E-HOSA: Extended-Hybrid Optical Switch Architecture for Cloud Computing Data Centers. In: International Conference on Frontiers of Information Technology (FIT), pp. 53–58.
    [25] Cisco Nexus 56128P Switch-Cisco. Available from: https://www.cisco.com/c/en/us/products/switches/nexus-56128p-switch/index.html.
    [26] Ye X, Mejia P, Yin Y, et al. (2010) DOS: a scalable optical switch for datacenters. In: Proceedings of the 6th ACM/IEEE Symposium on Architectures for Networking and Communications Systems, ACM.
    [27] Xi K, Kao YH and Chao HJ (2013) A petabit bufferless optical switch for data center networks. Optical Interconnects for Future Data Center Networks, pp. 135–154, Springer, New York, NY.
    [28] Xu M, Liu C and Subramaniam S (2018) PODCA: A Passive Optical Data Center Network Architecture. Journal of Optical Communications and Networking 10: 409–420. doi: 10.1364/JOCN.10.000409
    [29] Lu Y, Gu H, Yu X, et al. (2018) NEST: Towards Extreme Scale Computing Systems. IEICE T Inf Syst E101–D: 2827–2830.
    [30] Nakagawa M, Masumoto K, Onda H, et al. (2018) Photonic Sub-Lambda Transport: An Energy-Efficient and Reliable Solution for Metro Networks. In: 2018 International Conference on Optical Network Design and Modeling (ONDM), pp. 166–171.
    [31] Wang J, Basu S, McArdle C, et al. (2015) Large-scale hybrid electronic/optical switching networks for datacenters and HPC systems. In: 2015 IEEE 4th International Conference on Cloud Networking (CloudNet), pp. 87–93.
    [32] Shukla V and Jain A (2018) Design and analysis of high speed optical routers for next generation data centre network. Journal of Engineering Research 6: 122–137.
    [33] Romagnoli M, Sorianello V, Midrio M, et al. (2018) Graphene-based integrated photonics for next-generation datacom and telecom. Nature Reviews Materials 3: 392–414. doi: 10.1038/s41578-018-0040-9
    [34] Oki E, Shiomoto K, Shimazaki D, et al. (2005) Dynamic multilayer routing schemes in GMPLS-based IP+ optical networks. IEEE Commun Mag 43: 108–114.
    [35] Viswanathan A, Feldman N, Wang Z, et al. (1998) Evolution of multiprotocol label switching. IEEE Commun Mag 36: 165–173. doi: 10.1109/35.668287
    [36] Munoz R, Casellas R, Martinez R, et al. (2014) PCE: What is it, how does it work and what are its limitations? J Lightwave Technol 32: 528–543. doi: 10.1109/JLT.2013.2276911
    [37] Kreutz D, Ramos FMV, Verissimo PJE, et al. (2015) Software-defined networking: A comprehensive survey. P IEEE 103: 14–76. doi: 10.1109/JPROC.2014.2371999
    [38] Openflow1.4. [Online]. Available from: https://www.opennetworking.org/images/stories/downloads/sdn-resources/onf-specifications/op enflow/openflow-spec-v1.4.0.pdf.
    [39] Li Y and Kilper DC (2018) Optical Physical Layer SDN [Invited]. Journal of Optical Communications and Networking 10: A110–A121. doi: 10.1364/JOCN.10.00A110
    [40] Lallas EN (2019) A Survey on All Optical Label Swapping Techniques: Comparison and Trends. Opt Switch Netw 31: 22–38. doi: 10.1016/j.osn.2018.08.002
    [41] Cui L, Yu FR and Yan Q (2016) When big data meets software-defined networking: SDN for big data and big data for SDN. IEEE Network 30: 58–65. doi: 10.1109/MNET.2016.7389832
    [42] Casellas R, Martinez R, Vilalta R, et al. (2018) Control, Management, and Orchestration of Optical Networks: Evolution, Trends, and Challenges. J Lightwave Technol 36: 1390–1402. doi: 10.1109/JLT.2018.2793464
    [43] Thyagaturu AS, Mercian A, McGarry MP, et al. (2016) Software Defined Optical Networks (SDONs): A Comprehensive Survey. IEEE Commun Surv Tut 18: 2738–2786. doi: 10.1109/COMST.2016.2586999
    [44] OpenROADM. [Online]. Available from: http://www.openroadm.org.
    [45] Zhao J, Subramaniam S and Brandt-Pearce M (2014) Intradomain and interdomain QoT-aware RWA for translucent optical networks. J Opt Commun Netw 6: 536–548. doi: 10.1364/JOCN.6.000536
    [46] SDN Architecture, Issue 1.1, ONF TR-521, Open Networking Foundation 2016.
    [47] Network functions virtualisation (NFV), Architectural framework, ETSI GS NFV 002 (V1.2.1), Dec. 2014.
    [48] Mestres A, Rodriguez-Natal A, Carner J, et al. (2017) Knowledge-defined networking. ACM SIGCOMM Comp Com 47: 2–10.
    [49] Lu W, Liang L, Kong B, et al. (2018) AI-Assisted Knowledge-Defined Network Orchestration for Energy-Efficient Datacenter Networks. IEEE Commun Mag.
    [50] Musumeci F, Rottondi C, Nag A, et al. (2019) An Overview on Application of Machine Learning Techniques in Optical Networks. IEEE Commun Surv Tut 21: 1383–1408. doi: 10.1109/COMST.2018.2880039
    [51] Yamanaka N, Okamoto S, Hirono M, et al. (2018) Application-Triggered Automatic Distributed Cloud/Network Resource Coordination by Optically Networked Inter/Intra Data Center [invited]. J Opt Commun Netw 10: B15–B24. doi: 10.1364/JOCN.10.000B15
    [52] Huang MF (2017) Architecture of OpenFlow based Software Defined Optical Label Swapping. US PATENT No. 9.537.598.
    [53] Walkowiak K, Wozniak M, Klinkowski M, et al. (2015) Optical Networks for Cost-Efficient and Scalable Provisioning of Big Data Traffic. International Journal of Parallel, Emergent and Distributed Systems 30: 15–28. doi: 10.1080/17445760.2014.924516
    [54] Wu Z, Li J, Zhu P, et al. (2015) Experimental Demonstration of Optical Labeled Superchannel Switching for Elastic Optical Network. Optical Fiber Communication Conference, Optical Society of America.
    [55] Dallaglio M, Giorgetti A, Sambo N, et al. (2015) Provisioning and Restoration With Sliceability in GMPLS-Based Elastic Optical Networks. J Opt Commun Netw 7: A309–A317. doi: 10.1364/JOCN.7.00A309
    [56] Yoshida Y, Maruta A, Kitayama K, et al. (2015) SDN-Based Network Orchestration of Variable-Capacity Optical Packet Switching Network Over Programmable Flexi-Grid Elastic Optical Path Network. J Lightwave Technol 33: 609–617. doi: 10.1109/JLT.2014.2351852
    [57] Yin Y, Liu L, Proietti R, et al. (2017) Software Defined Elastic Optical Networks for Cloud Computing. IEEE Network 31: 4–10. doi: 10.1109/MNET.2017.8120233
    [58] Kanj M, Rouzic EL, Meuric J, et al. (2016) Optical power control in GMPLS control plane. IEEE/OSA Journal of Optical Communications and Networking 8: 553–568. doi: 10.1364/JOCN.8.000553
    [59] Kanj M, Rouzic EL, Meuric J, et al. (2018) Optical power control in translucent flexible optical networks with GMPLS control plane. IEEE/OSA Journal of Optical Communications and Networking 10: 760–772. doi: 10.1364/JOCN.10.000760
    [60] Blumenthal DJ, Bowers JE, Rau L, et al. (2003) Optical signal processing for optical packet switching networks. IEEE Commun Mag 41: S23–S29. doi: 10.1109/MCOM.2003.1179494
    [61] Guillemot C, Renaud M, Gambini P, et al. (1998) Transparent optical packet switching: The European ACTS KEOPS project approach. J Lightwave Technol 16: 2117–2134. doi: 10.1109/50.736580
    [62] Okada A (2002) All-optical packet routing in AWG-based wavelength routing networks using an out-of-band optical label. In: Optical Fiber Communication Conference and Exhibit, pp. 213–215.
    [63] Skoufis C, Sygletos S, Leligou N, et al. (2003) Data-Centric Networking Using Multiwavelength Headers/Labels in Packet-Over-WDM Networks:A Comparative Study. J Lightwave Technol 21: 2110–2122. doi: 10.1109/JLT.2003.816897
    [64] Zhu Z, Hernandez VJ, Jeon MY, et al. (2003) RF Photonics Signal Processing in Subcarrier Multiplexed Optical-Label Switching Communication Systems. J Lightwave Technol 21: 3155–3166. doi: 10.1109/JLT.2003.822237
    [65] Lallas EN, Skarmoutsos N and Syvridis D (2005) Coherent encoding of optical FSK header for All optical Label Swapping Systems. J Lightwave Technol 23: 1199–1209. doi: 10.1109/JLT.2005.843490
    [66] Chow CW, Wong CS and Tsang HK (2004) All-Optical ASK/DPSK Label-Swapping and Buffering Using Fabry–Perot Laser Diodes. IEEE J Sel Top Quant 10: 363–370. doi: 10.1109/JSTQE.2004.825955
    [67] Fouli K and Maier M (2007) OCDMA and optical coding: Principles, applications, and challenges. IEEE Commun Mag 45: 27–34.
    [68] Fakih A, Panbude S and Jagtap S (2014) Performance Analysis of Two Dimensional Wavelength/Time Encoding System for Optical CDMA Networks. International Journal of Computer and Communication Engineering 3: 424–428. doi: 10.7763/IJCCE.2014.V3.362
    [69] Matsumoto R, Kodama T, Morita K, et al. (2015) Scalable two- and three-dimensional optical labels generated by 128-port encoder/decoder for optical packet switching. Opt Express 23: 25747–25761. doi: 10.1364/OE.23.025747
    [70] Habib C, Baby V, Chen LR, et al. (2008) All-Optical Swapping of Spectral Amplitude Code Labels Using Nonlinear Media and Semiconductor Fiber Ring Lasers. IEEE J Sel Top Quant 14: 879–888. doi: 10.1109/JSTQE.2008.918047
    [71] Aboagye IA, Chen F and Cao Y (2017) Performance Analysis of 112 Gb/s×4-Channel WDM PDM-DQPSK Optical Label Switching System With Spectral Amplitude Code Labels. Photonic Sens 7: 88–96. doi: 10.1007/s13320-016-0345-5
    [72] Seddighian P, Ayotte S, Rosas-Fernández JB, et al. (2007) Label Stacking in Photonic Packet-Switched Networks With Spectral Amplitude Code Labels. J Lightwave Technol 25: 463–471. doi: 10.1109/JLT.2006.888265
    [73] Mendinueta JMD, Shinada S, Furukawa H, et al. (2017) Ultra-High-Capacity Optical Packet Switching Networks with Coherent Polarization Division Multiplexing QPSK/16QAM Modulation Formats. Photonics 4: 27. doi: 10.3390/photonics4020027
    [74] Eiselt M, Dochhan A, Elbers JP (2018) Data Center Interconnects at 400G and Beyond. ArXiv:1807.11861.
    [75] Moura PM, Scaraficci RA and da Fonseca NLS (2015) Algorithm for energy efficient routing, modulation and spectrum assignment. In: 2015 IEEE International Conference on Communications (ICC), pp. 5961–5966.
    [76] Xu Y, Li X and Yu J (2016) Simple scheme for PDM-QPSK payload generation in an optical label switching network. J Opt Commun Netw 8: 53–57. doi: 10.1364/JOCN.8.000053
    [77] Okonkwo CM, van Uden RGH, Chen H, et al. (2015) Advanced coding techniques for few mode transmission systems. Opt Express 23: 1411–1420. doi: 10.1364/OE.23.001411
    [78] Tarokh V, Jafarkhani H and Calderbank AR (1999) Space-Time Block Codes from Orthogonal Designs. IEEE T Inform Theory 45: 1456–1467. doi: 10.1109/18.771146
    [79] Alvarado A and Agrell E (2015) Four-Dimensional Coded Modulation with Bit-Wise Decoders for Future Optical Communications. J Lightwave Technol 33: 1993–2003. doi: 10.1109/JLT.2015.2396118
    [80] Isaac AA, Chen F, Cao Y, et al. (2017) Effects of Polarization Tracker on 80 and 112Gb/s PDM-DQPSK with Spectral Amplitude Code Labels. Proceeding of the World Congress on Engineering 1.
    [81] Nakagawa G, Feuer MD, Mikhailov V, et al. (2015) High-speed Polarization Shift Keying Lightpath Labeling of 100 Gb/s DP-QPSK for Programmable Photonic Networks. Optical Fiber Communication Conference, pp. Tu3D-2. Optical Society of America.
    [82] Xu M, Li Y, Kang TZ, et al. (2016) Performance evaluations of hybrid modulation with different optical labels over PDQ in high bit-rate OLS network systems. Opt Express 24: 26228–26240. doi: 10.1364/OE.24.026228
    [83] Khlifi Y and Alotaibi M (2018) A novel multicast grooming scheme for dynamic QoS provision over OLS networks. IEEE 8th Annual Computing and Communication Workshop and Conference (CCWC), pp. 911–917.
  • This article has been cited by:

    1. Timotei István Erdei, Rudolf Krakó, Géza Husi, Design of a Digital Twin Training Centre for an Industrial Robot Arm, 2022, 12, 2076-3417, 8862, 10.3390/app12178862
    2. Cheng-Long Wang, Shasha Gao, Xue-Zhi Li, Maia Martcheva, Modeling Syphilis and HIV Coinfection: A Case Study in the USA, 2023, 85, 0092-8240, 10.1007/s11538-023-01123-w
    3. Yujie Sheng, Jing-An Cui, Songbai Guo, The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy, 2023, 20, 1551-0018, 5966, 10.3934/mbe.2023258
    4. Hao Wang, Di Zhu, Shiqi Li, Robert A. Cheke, Sanyi Tang, Weike Zhou, Home quarantine or centralized quarantine? A mathematical modelling study on the COVID-19 epidemic in Guangzhou in 2021, 2022, 19, 1551-0018, 9060, 10.3934/mbe.2022421
    5. Israel Edem Agbehadji, Bankole Osita Awuzie, Alfred Beati Ngowi, COVID-19 Pandemic Waves: 4IR Technology Utilisation in Multi-Sector Economy, 2021, 13, 2071-1050, 10168, 10.3390/su131810168
    6. Tianheng Xie, Jianfang Zhang, Xiangtao Li, Data-Driven Intelligent Risk System in the Process of Financial Audit, 2022, 2022, 1563-5147, 1, 10.1155/2022/9054209
    7. Katica Tomic, 2023, Chapter 11, 978-3-031-13752-5, 301, 10.1007/978-3-031-13753-2_11
    8. Shasha Gao, Mingwang Shen, Xueying Wang, Jin Wang, Maia Martcheva, Libin Rong, A multi-strain model with asymptomatic transmission: Application to COVID-19 in the US, 2023, 565, 00225193, 111468, 10.1016/j.jtbi.2023.111468
    9. 德玉 郭, A Dynamics Model of COVID-19 with Transition from the Asymptomatic to the Symptomatic Infected Individuals, 2023, 12, 2324-7991, 2457, 10.12677/AAM.2023.125248
    10. Kaijing Chen, Fengying Wei, Xinyan Zhang, Hao Jin, Zuwen Wang, Yue Zuo, Kai Fan, Epidemiological feature analysis of SVEIR model with control strategy and variant evolution, 2024, 9, 24680427, 689, 10.1016/j.idm.2024.03.005
    11. Shasha Gao, Pant Binod, Chidozie Williams Chukwu, Theophilus Kwofie, Salman Safdar, Lora Newman, Seoyun Choe, Bimal Kumar Datta, Wisdom Kwame Attipoe, Wenjing Zhang, P. van den Driessche, A mathematical model to assess the impact of testing and isolation compliance on the transmission of COVID-19, 2023, 8, 24680427, 427, 10.1016/j.idm.2023.04.005
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5680) PDF downloads(1619) Cited by(4)

Figures and Tables

Figures(8)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog