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A generalized n-site Asymmetric Simple Inclusion Process (ASIP) network is studied, where gate-opening instants are determined by a renewal process and arrivals occur to all sites. Various types of batch particle movements between sites are analyzed: (i) unidirectional probabilistic forward movements; (ii) probabilistic forward movements combined with feedback to the first site; and (iii) general probabilistic multidirectional movements. In contrast to the tedious successive substitution method used in previous ASIP studies, an efficient matrix approach is applied to derive the multidimensional probability generating function (PGF) of site occupancies right after gate opening instants. The complexity of the ASIP processes allows us to obtain explicit PGF results for small-size networks only, while for larger networks, a formula to calculate the mean site occupancies is derived for all types of movements. In movement case (i) the means are directly and explicitly calculated. For movement case (ii), where the network is homogeneous with equal probabilities of forward movements from site i to downstream sites j ≥ i , we show that the ratio between the mean occupancies of consecutive sites approaches a constant when the network becomes large, and calculate this ratio. Finally, we investigate an n-site network where at gate opening instants all gates open simultaneously, and particles move in all directions. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

In various real-life queueing systems, part of the service can be rendered without involvement or presence of the customers themselves. In those queues, customers whose service order is still in process may leave the service station, go to ‘orbit’ for a random length of time, and then return to find out if their order has been completed. Common examples are car’s annual maintenance works, food ordering, etc. In this paper, a thorough analysis of a single-server ‘orbit while in service’ queueing model with general service time is presented. Assuming an Exponentially distributed orbit time, we derive general formulae for the distributions of (i) a customer’s total residence time in the system; (ii) a customer’s net actual residence time in the system during service (not including orbit time); (iii) the time an orbiting customer is late to return, i.e., remains in orbit after his/her service has been completed; and (iv) the total number of customers in the system. Considering the family of Gamma-distributed service times (spanning the range of distributions between the Exponential and the Deterministic), as well as the Uniform distribution, we further derive explicit formulae for the distributions of the above variables. Under linear cost assumptions, the optimal mean orbit time is numerically calculated for each of the above service-time distributions. Figures depicting the behavior of the measures as functions of the parameters are presented. [ABSTRACT FROM AUTHOR]

A 2-queue system with a single-server operating according to the combined 'Join the Shortest Queue–Serve the Longest Queue' regime is analyzed. Both cases, with or without server's switch-over times, are investigated under the non-preemptive discipline. Instead of dealing with a state space comprised of two un-bounded dimensions, a non-conventional formulation is constructed, leading to an alternative two-dimensional state space, where only one dimension is infinite. As a result, the system is defined as a quasi birth and death process and is analyzed via both the probability generating functions method and the matrix geometric formulation. Consequently, the system's two-dimensional probability mass function is derived, from which the system's performance measures, such as mean queue sizes, mean sojourn times, fraction of time the server resides in each queue, correlation coefficient between the queue sizes, and the probability mass function of the difference between the queue sizes, are obtained. Extensive numerical results for various values of the system's parameters are presented, as well as a comparison between the current non-preemptive model and its twin system of preemptive service regime. One of the conclusions is that, depending on the variability of the various parameters, the preemptive regime is not necessarily more efficient than the non-preemptive one. Finally, economic issues are discussed and numerical comparisons are presented, showing the advantages and disadvantages of each regime. [ABSTRACT FROM AUTHOR]

A double-space parking problem is studied for a parking lot of size M accommodating both private cars and buses. Upon arrival, a private car is either admitted to the parking lot, occupying a single spot, or waits in line until a spot becomes available. An arriving bus occupies double spots and is admitted only if there are at least two free spots. It balks from the system otherwise. The inflow is governed by two independent Poisson streams, with rates λ C for cars and λ B for buses. The sojourn time of a car or a bus inside the parking lot is exponentially distributed with parameters μ C and μ B , respectively. The problem is formulated as a QBD process and analyzed via matrix geometric methods. Various performance measures are calculated, including mean number of cars inside, and outside, the parking lot; mean number of buses in the system; and the probability that an arriving bus is blocked. The dichotomy whether to split the M-spot lot into two separate lots, one for cars, the other for buses, is studied and the optimal split is calculated. Numerical results are presented via graphs. Finally, it is shown that from the point of view of the parking lot owner, it is equivalent to either charge a fixed entrance fee or charge per-time unit of usage. [ABSTRACT FROM AUTHOR]

We consider a single-server multi-queue system with unlimited-size batch service where the next queue to be served is the one with the most senior customer (the so called 'Israeli Queue'). We study a Markovian system with state-dependent group-joining policy and derive results for various performance measures, such as steady-state distribution of the number of groups in the system, sojourn times, group sizes, and lengths of busy periods. Closed-form expressions are obtained for both the Uniform and the Geometric joining policies. Numerical results are presented. [ABSTRACT FROM AUTHOR]

The workload of a generalized n-site asymmetric simple inclusion process (ASIP) is investigated. Three models are analyzed. The first model is a serial network for which the steady-state Laplace–Stieltjes transform (LST) of the total workload in the first k sites ( k ≤ n ) just after gate openings and at arbitrary epochs is derived. In a special case, the former (just after gate openings) turns out to be an LST of the sum of k independent random variables. The second model is a 2-site ASIP with leakage from the first queue. Gate openings occur at exponentially distributed intervals, and the external input processes to the stations are two independent subordinator Lévy processes. The steady-state joint workload distribution right after gate openings, right before gate openings and at arbitrary epochs is derived. The third model is a shot-noise counterpart of the second model where the workload at the first queue behaves like a shot-noise process. The steady-state total amount of work just before a gate opening turns out to be a sum of two independent random variables. [ABSTRACT FROM AUTHOR]

Clearly, HT ht [[4]]. Suppose customer I j i arrives at time HT ht and departs at time HT ht while requiring a total service time HT ht . The overall discrimination of this customer, HT ht , is the difference between the customer's service requirement and the total attained service during the customer's sojourn time in the system: HT ht , where HT ht might be positive or negative. [Extracted from the article]

This paper considers an unobservable two-site tandem queueing system attended by an alternating server. We study the strategic customer behaviour under two threshold-based operating policies, applied by a profit-maximizing server, while customers' strategic behaviour and server's switching costs are taken into account. Under the Exact-N policy, in each cycle the server first completes service of N customers in the first stage ( Q 1 ), then switches to the second stage ( Q 2 ) and then serves those N customers before switching back to Q 1 to start a new cycle. This policy leads to a mixture of Follow-the-Crowd and Avoid-the-Crowd customer behaviour. In contrast, under the N-Limited policy, the server switches from Q 1 to Q 2 also when the first queue is emptied, making this regime work-conserving and leading only to Avoid-the-Crowd behaviour. Performance measures are obtained using matrix geometric methods for both policies and any threshold N, while for sequential service ( N = 1 ) explicit expressions are derived. It is shown that the system's stability condition is independent of N, and of the switching policy. Optimal performances in equilibrium, under each of these switching policies, are analysed and compared through a numerical study. [ABSTRACT FROM AUTHOR]

We study a Markovian single-server ticket queue where, upon arrival, each customer can draw a number from a take-a-number machine, while the number of the customer currently being served is displayed on a panel. The difference between the above two numbers is called the "virtual queue length." We consider a nonhomogeneous population of customers comprised of two types: "regular" and "strategic." Upon arrival, a regular customer, regardless of the value of the virtual queue length, draws a number from the machine, joins the queue and waits in the system until being served. A strategic customer, depending on the virtual queue length, may either join, leave, or go to "orbit" for a random duration. If, upon return from orbit, a strategic customer realizes that s/he missed her/his turn, s/he balks. Otherwise, s/he joins the queue and waits to be served. We analyze this intricate stochastic system, calculate its steady-state probabilities, derive the sojourn time's Laplace–Stieltjes transform of a regular and of a strategic customer and calculate the system's performance measures. Finally, an economic analysis is performed to determine the optimal mean orbiting time of strategic customers for two types of objective functions. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

Exact analysis of a multi-server Markovian queueing system with cross selling in steady-state is presented. Cross selling attempt is initiated at the end of a customer's service every time the number of customers in the system is below a threshold. Both probability generating functions (PGFs) and matrix geometric methods are employed. The relation between the methods is revealed by explicitly calculating the entries of the matrix geometric rate-matrix R. Those entries are expressed in terms of the roots of a determinant of a matrix related to the set of linear equations involving the PGFs. This is a further step towards understanding of the analytical relationship between the two methods. Numerical results are presented, showing the effect of the cross selling intensity and of the threshold level on the systems performance measures. Finally, for a given set of parameters, the optimal threshold level is determined. [ABSTRACT FROM AUTHOR]

We consider a system of two separate finite-buffer M / M / 1 queues served by a single server, where the switching mechanism between the queues is threshold-based, determined by the queue which is not being served. Applications may be found in data centers, smart traffic-light control and human behavior. Specifically, whenever the server attends queue i ( $$Q_i$$ ) and the number of customers in the other queue, $$Q_j$$ ( $$i,j=1,2$$ ; $$j\ne i$$ ), reaches its threshold level, the server immediately switches to $$Q_j$$ whenever $$Q_i$$ is below its threshold. When a served $$Q_i$$ becomes empty we consider two scenarios: (i) non-work-conserving; and (ii) work-conserving. We present occasions where the non-work-conserving policy is more economical than the work-conserving policy when high switching costs are involved. An intrinsic feature of the process is an oscillation phenomenon: when the occupancy of $$Q_i$$ decreases the occupancy of the other queue increases. This fact is illustrated and discussed. By formulating the system as a three-dimensional continuous-time Markov chain we provide a probabilistic analysis of the system and investigate the effects of buffer sizes and arrival rates, as well as service rates, on the system's performance. Numerical examples are presented and extreme cases are investigated. [ABSTRACT FROM AUTHOR]

We consider an asymmetric inclusion process, which can also be viewed as a model of n queues in series. Each queue has a gate behind it, which can be seen as a server. When a gate opens, all customers in the corresponding queue instantaneously move to the next queue and form a cluster with the customers there. When the nth gate opens, all customers in the nth site leave the system. For the case where the gate openings are determined by a Markov renewal process, and for a quite general arrival process of customers at the various queues during intervals between successive gate openings, we obtain the following results: (i) steady-state distribution of the total number of customers in the first k queues, $$k=1,\dots ,n$$ ; (ii) steady-state joint queue length distributions for the two-queue case. In addition to the case that the numbers of arrivals in successive gate opening intervals are independent, we also obtain explicit results for a two-queue model with renewal arrivals. [ABSTRACT FROM AUTHOR]

The so-called 'Israeli queue' (Boxma et al. in Stoch Model 24(4):604-625, ; Perel and Yechiali in Probab Eng Inf Sci, ; Perel and Yechiali in Stoch Model 29(3):353-379, ) is a multi-queue polling-type system with a single server. Service is given in batches, where the batch sizes are unlimited and the service time of a batch does not depend on its size. After completing service, the next queue to be visited by the server is the one with the most senior customer. In this paper, we study the Israeli queue with retrials, where the system is comprised of a 'main' queue and an orbit queue. The main queue consists of at most $$M$$ groups, where a new arrival enters the main queue either by joining one of the existing groups, or by creating a new group. If an arrival cannot join one of the groups in the main queue, he goes to a retrial (orbit) queue. The orbit queue dispatches orbiting customers back to the main queue at a constant rate. We analyze the system via both probability generating functions and matrix geometric methods, and calculate analytically various performance measures and present numerical results. [ABSTRACT FROM AUTHOR]

Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type- $$i$$ job ( $$i=1,2$$ ) finds the server busy, it is blocked and routed to a separate type- $$i$$ retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a Riemann-Hilbert boundary value problem. Various performance measures are calculated and numerical results are presented. In particular, numerical results demonstrate that the proposed multiple access system with two types of jobs and constant retrial rates provides incentives for the users to respect their contracts. [ABSTRACT FROM AUTHOR]

We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Markovian system with blocking at the second queue, analyze it, and derive its stability condition. We then study a non-Markovian setting and derive the stability condition for an approximating diffusion, showing its similarity to the former condition. In the Markovian setting, we use probability generating functions and matrix analytic techniques. In the diffusion setting, we consider expectations of the first hitting times of compact sets. [ABSTRACT FROM AUTHOR]

We study N-queues single-server fluid polling systems, where a fluid is continuously flowing into the queues at queue-dependent rates. When visiting and serving a queue, the server reduces the amount of fluid in the queue at a queue-dependent rate. Switching from queue i to queue j requires two random-duration steps: (i) departing queue i, and (ii) reaching queue j. The length of time the server resides in a queue depends on the service regime. We consider three main regimes: Exhaustive, Gated, and Globally-Gated. Two polling procedures are analyzed: (i) cyclic and (ii) probabilistic. Under steady-state, we derive the Laplace–Stieltjes transform (LST), mean, and second moment of the amount of flow at each queue at polling instants, as well as at an arbitrary moment. We further calculate the LST and mean of the “waiting time” of a drop at each queue and derive expressions for the mean total load in the system for the various service regimes. Finally, we explore optimal switching procedures. [ABSTRACT FROM AUTHOR]

We consider a system comprised of two connected M/ M/•/• type queues, where customers of one queue act as servers for the other queue. One queue, Q 1, operates as a limited-buffer M/ M/1/ N−1 system. The other queue, Q 2, has an unlimited-buffer and receives service from the customers of Q 1. Such analytic models may represent applications like SETI@home, where idle computers of users are used to process data collected by space radio telescopes. Let L 1 denote the number of customers in Q 1. Then, two models are studied, distinguished by their service discipline in Q 2: In Model 1, Q 2 operates as an unlimited-buffer, single-server M/ M/1/∞ queue with Poisson arrival rate λ 2 and dynamically changing service rate μ 2 L 1. In Model 2, Q 2 operates as a multi-server M/ M/ L 1/∞ queue with varying number of servers, L 1, each serving at a Poisson rate of μ 2. We analyze both models and derive the Probability Generating Functions of the system’s steady-state probabilities. We then calculate the mean total number of customers present in each queue. Extreme cases are indicated. [ABSTRACT FROM AUTHOR]

We consider a Jackson-type network comprised of two queues having state-dependent service rates, in which the queue lengths evolve periodically, exhibiting noisy cycles. To reduce this noise a certain heuristic, utilizing regions in the phase space in which the system behaves almost deterministically, is applied. Using this heuristic, we show that in order to decrease the probability of a customers overflow in one of the queues in the network, the server in that same queue – contrary to intuition – should be shut down for a short period of time. Further noise reduction is obtained if the server in the second queue is briefly shut down as well, when certain conditions hold. [ABSTRACT FROM AUTHOR]

Abstract. We derive performance measures for burst arrival (e.g. messages of variable length packets) M[sup K]/G/1 queues with server vacations, controlled by the, so called, Randomly Timed Gated (RTG) protocol, operating as follows: Whenever the server returns from a (general-type) vacation and initiates a busy period, a Timer with random duration T is activated. If the server empties the queue before time T, he leaves for another vacation. Otherwise (i.e. if there are still customers (packets) in the system when the timer expires), two versions of terminating the busy period, each leading to a different model, are studied: (i). The server completes service (e.g. transmission) only to the customer being served at time T and leaves. (ii). The server leaves immediately. We derive both state-dependent and steady-state performance measures as a function of the (vacation-type dependent variable) number of customers present at the initiation of a busy period. When the vacation policy is specified (i.e. Multiple or Single), we obtain explicit formulas for the various performance measures, derive the distribution and mean of the waiting and sojourn times of a customer, and compare between the two versions. Analysis of the conditions for stability concludes the paper. [ABSTRACT FROM AUTHOR]

Closed polling systems with station breakdowns, under the gated, exhaustive or globally gated services regimes, are studied and analyzed. Multi-dimensional sets of probability generating functions of the system's state are derived. They are further utilized to obtain an approximate solution for the mean number of jobs residing in the system's various queues at polling instants. The analysis is then concentrated on the case of cyclic Bernoulli polling. Explicit formulae for the mean number of jobs, as well as for the expected cycle duration and system utilization, are derived. Comparison of the throughputs of the three regimes concludes the paper. [ABSTRACT FROM AUTHOR]

We study two new cyclic reservation schemes for the efficient operation of systems consisting of a single server and multiple queues. The schemes are the Globally Gated regime and the Cyclic-Reservation Multiple-Access (CRMA). Both procedures possess mechanisms for prioritizing the queues and lend themselves to a closed-form analysis. The combination of these two properties allows for effective and efficient operation of the systems, for which we provide a thorough delay analysis and derive simple rules for optimal operation. [ABSTRACT FROM AUTHOR]

We introduce, analyse and optimize the class of Bernoulli random polling systems. The server moves cyclically among N channels (queues), but Change-over times between stations are composed of walking times required to 'move' from one channel to another and switch-in times that are incurred only when the server actually enters a station to render service. The server uses a Bernoulli random mechanism to decide whether to serve a queue or not: upon arrival to channel i, it switches in with probability p, or moves on to the next queue (w.p. 1 - p) without serving any customer (e.g. packet or job). The Cyclic Bernoulli Polling (CBP) scheme is independent of the service regime in any particular station, and may be applied to any service discipline. In this paper we analyse three different service disciplines under the CBP scheme: Gated, Partially Exhaustive and Fully Exhaustive. For each regime we derive expressions for (i) the generating functions and moments of the number of customers (jobs) at the various queues at polling instants, (ii) the expected number of jobs that an arbitrary departing job leaves behind it, and (iii) the LST and expectation of the waiting time of a cutomer at any given queue. The fact that these measures of performance can be explicitly obtained under the CBP is an advantage over all 'parameterized' cyclic polling schemes (such as the k-limited discipline) that have been studied in the literature, and for which explicit measures of performance are hard to obtain. The choice of the p's in the CBP allows for fine tuning and optimization of performance measures, as well as prioritization between stations (this being achieved at a low computational cost). For this purpose, we develop a Pseudo-conservation law for a mixed system comprised of channels from all three service disciplines, and define a Mathematical Program to find the optimal values of the probabilities { p}=1 so as to minimize the expected amount of unfinished work in the system. Any CBP scheme for which the optimal p's are not all equal to one, yields a smaller amount of the expected unfinished work in the system than that in the standard cyclic polling procedure with equivalent parameters. We conclude by showing that even in the case of a single queue, it is not always true that p=1 is the best strategy, and derive conditions under which it is optimal to have p < 1. [ABSTRACT FROM AUTHOR]

We study the behavior of a single-server discrete-time queue with batch arrivals, where the information on the queue length and possibly on service completions is delayed. Such a model describes situations arising in high speed telecommunication systems, where information arrives in messages, each comprising a variable number of fixed-length packets, and it takes one unit of time (a slot) to transmit a packet. Since it is not desirable to attempt service when the system may be empty, we study a model where we assume that service is attempted only if, given the information available to the server, it is certain that there are messages in the queue. We characterize the probability distribution of the number of messages in the queue under some general stationarity assumptions on the arrival process, when information on the queue size is delayed K slots, and derive explicit expressions of the PGF of the queue length for the case of i.i.d. batch arrivals and general independent service times. We further derive the PGF of the queue size when information on both the queue length and service completion is delayed K=1 units of time. Finally, we extend the results to priority queues and show that when all messages are of unit length, the cμ rule remains optimal even in the case of delayed information. [ABSTRACT FROM AUTHOR]

We consider a polling system consisting of N queues and a single server where polling is performed according to an Elevator (scan) scheme. The server first serves queues in the 'up' direction, i.e. in the order 1, 2,..., N − 1, N, and then serves these queues in the opposite ('down') direction, i.e. visiting them in the order N,N−1,...,2,1. The server then changes direction again, and so on. A globally gating regime is used each time the server changes direction. We show that, for this Elevator scheme, the expected waiting times in all channels are equal. This is the only known non-symmetric polling system that exhibits such a fairness phenomenon. We then discuss the problem of optimally ordering the queues so as to minimize some measure of variability of the waiting times. [ABSTRACT FROM AUTHOR]

An N-node tandem queueing network with Bernoulli feedback to the end of the queue of the first node is considered. We first revisit the single-node M/G/1 queue with Bernoulli feedback, and derive a formula for EL(n), the expected queue length seen by a customer at his nth feedback. We show that, as n becomes large, EL(n) tends to ρ/(l ρ), ρ being the effective traffic intensity. We then treat the entire queueing network and calculate the mean value of S, the total sojourn time of a customer in the N-node system. Based on these results we study the problem of optimally ordering the nodes so as to minimize ES. We show that this is a special case of a general sequencing problem and derive sufficient conditions for an optimal ordering. A few extensions of the serial queueing model are also analyzed. We conclude with an appendix in which we derive an explicit formula for the correlation coefficient between the number of customers seen by an arbitrary arrival to an M/G/1 queue, and the number of customers he leaves behind him upon departure. For the M/M/1 queue this coefficient simply equals the traffic intensity ρ. [ABSTRACT FROM AUTHOR]