Numerical methods for queues with shared service

A queueing system is a mathematical abstraction of a situation where elements, called customers, arrive in a system and wait until they receive some kind of service. Queueing systems are omnipresent in real life. Prime examples include people waiting at a counter to be served, airplanes waiting to take off, traffic jams during rush hour etc. Queueing theory is the mathematical study of queueing phenomena. As often neither the arrival instants of the customers nor their service times are known in advance, queueing theory most often assumes that these processes are random variables. The queueing process itself is then a stochastic process and most often also a Markov process, provided a proper description of the state of the queueing process is introduced. This dissertation investigates numerical methods for a particular type of Markovian queueing systems, namely queueing systems with shared service. These queueing systems differ from traditional queueing systems in that there is simultaneous service of the head-of-line customers of all queues and in that there is no service if there are no customers in one of the queues. The absence of service whenever one of the queues is empty yields particular dynamics which are not found in traditional queueing systems. These queueing systems with shared service are not only beautiful mathematical objects in their own right, but are also motivated by an extensive range of applications. The original motivation for studying queueing systems with shared service came from a particular process in inventory management called kitting. A kitting process collects the necessary parts for an end product in a box prior to sending it to the assembly area. The parts and their inventories being the customers and queues, we get ``shared service'' as kitting cannot proceed if some parts are absent. Still in the area of inventory management, the decoupling inventory of a hybrid make-to-stock/make-to-order system exhibits shared service. The production process prior to the decoupling inventory is make-to-stock and driven by demand forecasts. In contrast, the production process after the decoupling inventory is make-to-order and driven by actual demand as items from the decoupling inventory are customised according to customer specifications. At the decoupling point, the decoupling inventory is complemented with a queue of outstanding orders. As customisation only starts when the decoupling inventory is nonempty and there is at least one order, there is again shared service. Moving to applications in telecommunications, shared service applies to energy harvesting sensor nodes. Such a sensor node scavenges energy from its environment to meet its energy expenditure or to prolong its lifetime. A rechargeable battery operates very much like a queue, customers being discretised as chunks of energy. As a sensor node requires both sensed data and energy for transmission, shared service can again be identified. In the Markovian framework, "solving" a queueing system corresponds to finding the steady-state solution of the Markov process that describes the queueing system at hand. Indeed, most performance measures of interest of the queueing system can be expressed in terms of the steady-state solution of the underlying Markov process. For a finite ergodic Markov process, the steady-state solution is the unique solution of $N-1$ balance equations complemented with the normalisation condition, $N$ being the size of the state space. For the queueing systems with shared service, the size of the state space of the Markov processes grows exponentially with the number of queues involved. Hence, even if only a moderate number of queues are considered, the size of the state space is huge. This is the state-space explosion problem. As direct solution methods for such Markov processes are computationally infeasible, this dissertation aims at exploiting structural properties of the Markov processes, as to speed up computation of the steady-state solution. The first property that can be exploited is sparsity of the generator matrix of the Markov process. Indeed, the number of events that can occur in any state --- or equivalently, the number of transitions to other states --- is far smaller than the size of the state space. This means that the generator matrix of the Markov process is mainly filled with zeroes. Iterative methods for sparse linear systems --- in particular the Krylov subspace solver GMRES --- were found to be computationally efficient for studying kitting processes only if the number of queues is limited. For more queues (or a larger state space), the methods cannot calculate the steady-state performance measures sufficiently fast. The applications related to the decoupling inventory and the energy harvesting sensor node involve only two queues. In this case, the generator matrix exhibits a homogene block-tridiagonal structure. Such Markov processes can be solved efficiently by means of matrix-geometric methods, both in the case that the process has finite size and --- even more efficiently --- in the case that it has an infinite size and a finite block size. Neither of the former exact solution methods allows for investigating systems with many queues. Therefore we developed an approximate numerical solution method, based on Maclaurin series expansions. Rather than focussing on structural properties of the Markov process for any parameter setting, the series expansion technique exploits structural properties of the Markov process when some parameter is sent to zero. For the queues with shared exponential service and the service rate sent to zero, the resulting process has a single absorbing state and the states can be ordered such that the generator matrix is upper-diagonal. In this case, the solution at zero is trivial and the calculation of the higher order terms in the series expansion around zero has a computational complexity proportional to the size of the state space. This is a case of regular perturbation of the parameter and contrasts to singular perturbation which is applied when the service times of the kitting process are phase-type distributed. For singular perturbation, the Markov process has no unique steady-state solution when the parameter is sent to zero. However, similar techniques still apply, albeit at a higher computational cost. Finally we note that the numerical series expansion technique is not limited to evaluating queues with shared service. Resembling shared queueing systems in that a Markov process with multidimensional state space is considered, it is shown that the regular series expansion technique can be applied on an epidemic model for opinion propagation in a social network. Interestingly, we find that the series expansion technique complements the usual fluid approach of the epidemic literature.

[1]  Sanjib Kumar Panda,et al.  Review of Energy Harvesting Technologies for Sustainable Wireless Sensor Network , 2012 .

[2]  Biplab Sikdar,et al.  Energy efficient transmission strategies for Body Sensor Networks with energy harvesting , 2008, 2008 42nd Annual Conference on Information Sciences and Systems.

[3]  Aylin Yener,et al.  Optimum Transmission Policies for Battery Limited Energy Harvesting Nodes , 2010, IEEE Transactions on Wireless Communications.

[4]  Mandyam M. Srinivasan,et al.  Random review production/inventory systems with compound Poisson demands and arbitrary processing times , 1991 .

[5]  Dieter Fiems,et al.  A Queueing Theoretic Approach to Decoupling Inventory , 2012, ASMTA.

[7]  Vinod Sharma,et al.  Optimal energy management policies for energy harvesting sensor nodes , 2008, IEEE Transactions on Wireless Communications.

[8]  Ye Xia,et al.  Maximizing the Lifetime of Wireless Sensor Networks with Mobile Sink in Delay-Tolerant Applications , 2010, IEEE Transactions on Mobile Computing.

[9]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[10]  Georg Heinecke,et al.  Hybrid Production Strategy Between Make-to-Order and Make-to-Stock – A Case Study at a Manufacturer of Agricultural Machinery with Volatile and Seasonal Demand , 2012 .

[11]  Masoud Rabbani,et al.  Capacity coordination in hybrid make-to-stock/make-to-order production environments , 2012 .

[12]  Biplab Sikdar,et al.  Performance Modeling of Transmission Schedulers for Sensor Networks Capable of Energy Harvesting , 2010, 2010 IEEE International Conference on Communications.

[13]  Weifa Liang,et al.  Delay-tolerant data gathering in energy harvesting sensor networks with a mobile sink , 2012, 2012 IEEE Global Communications Conference (GLOBECOM).

[14]  Mandyam M. Srinivasan,et al.  The continuous review (s,S) policy for production/inventory systems with Poisson demands and arbitrary processing times , 1987 .

[15]  Levent Gun,et al.  Experimental results on matrix-analytical solution techniques–extensions and comparisons , 1989 .

[16]  S. Hoekstra,et al.  Integral Logistic Structures: Developing Customer-Oriented Goods Flow , 1992 .

[17]  Mandyam M. Srinivasan,et al.  The (s,S) policy for the production/inventory system with compound Poisson demands , 1988 .

[18]  Aliakbar Montazer Haghighi,et al.  Queueing Models in Industry and Business , 2014 .

[19]  Damla Turgut,et al.  Heuristic Approaches for Transmission Scheduling in Sensor Networks with Multiple Mobile Sinks , 2011, Comput. J..

[20]  Herwig Bruneel,et al.  Analytic study of the queueing performance and the departure process of a leaky bucket with bursty input traffic. , 1996 .

[21]  Peter Buchholz,et al.  An EM-Algorithm for MAP Fitting from Real Traffic Data , 2003, Computer Performance Evaluation / TOOLS.

[22]  Purushottam Kulkarni,et al.  Energy Harvesting Sensor Nodes: Survey and Implications , 2011, IEEE Communications Surveys & Tutorials.

[23]  Mani B. Srivastava,et al.  Emerging techniques for long lived wireless sensor networks , 2006, IEEE Communications Magazine.

[24]  Nipa Phojanamongkolkij,et al.  A hybrid push/pull system in assemble-to-order manufacturing environment , 2009, J. Intell. Manuf..

[25]  Onur Kaya,et al.  Combined make-to-order/make-to-stock supply chains , 2009 .

[26]  Herwig Bruneel,et al.  A heuristic analytic technique to calculate the cell loss ratio in a leaky bucket with bursty input traffic , 1994 .

[27]  D. P. Donk,et al.  Combined make-to-order and make-to-stock in a food production system , 2004 .

[28]  Joan Ventura,et al.  Markov modeling of energy harvesting Body Sensor Networks , 2011, 2011 IEEE 22nd International Symposium on Personal, Indoor and Mobile Radio Communications.

[29]  Masoud Rabbani,et al.  An MADM Framework toward Hierarchical Production Planning in Hybrid MTS/MTO Environments , 2009 .

[30]  Abdur Rahim,et al.  Optimal production-inventory policy for make-to-order versus make-to-stock based on the M/Er/1 queuing model , 2007 .

[31]  Jing Yang,et al.  Optimal Packet Scheduling in an Energy Harvesting Communication System , 2010, IEEE Transactions on Communications.

[32]  Evgenia Smirni,et al.  KPC-Toolbox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes , 2008, 2008 Fifth International Conference on Quantitative Evaluation of Systems.

[33]  Christina Nielsen,et al.  An analytical study of the Q , 2005, Eur. J. Oper. Res..

[34]  Mahmut Parlar,et al.  Continuous-review inventory problem with random supply interruptions , 1997 .

[35]  Gregory A. DeCroix,et al.  Make-to-order versus make-to-stock in a production-inventory system with general production times , 1998 .

[36]  Ivo J. B. F. Adan,et al.  Combining make to order and make to stock , 1998 .

[37]  Hwee Pink Tan,et al.  Energy-Aware Transmission Control for Wireless Sensor Networks Powered by Ambient Energy Harvesting: A Game-Theoretic Approach , 2011, 2011 IEEE International Conference on Communications (ICC).