A product form solution to a system with multi-type jobs and multi-type servers

We consider a memoryless single station service system with servers \(\mathcal{S}=\{m_{1},\ldots,m_{K}\}\), and with job types \(\mathcal{C}=\{a,b,\ldots\}\). Service is skill-based, so that server m _i can serve a subset of job types \(\mathcal{C}(m_{i})\). Waiting jobs are served on a first-come-first-served basis, while arriving jobs that find several idle servers are assigned to a feasible server randomly. We show that there exist assignment probabilities under which the system has a product-form stationary distribution, and obtain explicit expressions for it. We also derive waiting time distributions in steady state.     

Choosing among heterogeneous server clouds

This paper considers a model of interest in cloud computing applications. We consider a multiserver system consisting of N heterogeneous servers. The servers are categorized into M(\(\ll N\)) different types according to their service capabilities. Jobs having specific resource requirements arrive at the system according to a Poisson process with rate \(N \lambda \). Upon each arrival, a small number of servers are sampled uniformly at random from each server type. The job is then routed to the sampled server with maximum vacancy per server capacity. If a job cannot obtain the required amount of resources from the server to which it is assigned, then the job is discarded. We analyze the system in the limit as \(N \rightarrow \infty \). This gives rise to a mean field, which we show has a unique fixed point and is globally attractive. Furthermore, as \(N\rightarrow \infty \), the servers behave independently. The stationary tail probabilities of server occupancies are obtained from the stationary solution of the mean field. Numerical results suggest that the proposed scheme significantly reduces the average blocking probability compared to static schemes that probabilistically route jobs to servers in proportion to the number of servers of each type. Moreover, the reduction in blocking holds even for systems at high load. For the limiting system in statistical equilibrium, our simulation results indicate that the occupancy distribution is insensitive to the holding time distribution and only depends on its mean.     

On the Relative Waiting Times in the GI/M/s and the GI/M/1 Queueing Systems

The expected steady-state waiting time, Wq(s), in a GI/M/s system with interarrival-time distribution H(·) is compared with the mean waiting time, Wq, in an "equivalent" system comprised of s separate GI/M/1 queues each fed by an interarrival-time distribution G(·) with mean arrival rate equal to 1/s times that of H(·). For H(·) assumed to be Exponential, Gamma or Deterministic three possible relationships between H(·) and G(·) are considered: G(·) can be of the "same type" as H(·); G(·) can be derived from H(·) by assigning new arrivals to the individual channels in a cyclic order; and G(·) may be obtained from H(·) by assigning customers probabilistically to the different queues. The limiting behaviour of the ratio R = Wq/Wq(s) is studied for the extreme values (1 and 0) of the common traffic intensity, ρ. Closed form results, which depend on the forms of H(·) and G(·) and on the relationships between them, are derived. It is shown that Wq is greater than Wq(s) by a factor of at least (s + 1)/2 when ρ approaches one, and that R is at least s(s!) when ρ tends to zero. In the latter case, however, R goes to infinity (!) in most cases treated. The results may be used to evaluate the effect on the waiting times when, for certain (non-queueing) reasons, it is needed to partition a group of s servers into several small groups.     

Refining Cosmetatos' Approximation for the Mean Waiting Time in the M/D/s Queue

This paper deals with refining Cosmetatos's approximation for the mean waiting time in an M/D/s queue. Although his approximation performs quite well in heavy traffic, it overestimates the true value when the number of servers is large or the traffic is light. We first focus on a normalized quantity that is a ratio of the mean waiting times for the M/D/s and M/M/s queues. Using some asymptotic properties of the quantity, we modify Cosmetatos's approximation to obtain better accuracy both for large s and in light traffic. To see the quality of our approximation, we compare it with the exact value and some previous approximations. Extensive numerical tests indicate that the relative percentage error is less than 1% for almost all cases with s ≤ 20 and at most 5% for other cases.     

A queueing model for crowdsourcing

Crowdsourcing is getting popular after a number of industries such as food, consumer products, hotels, electronics, and other large retailers bought into this idea of serving customers. In this paper, we introduce a multi-server queueing model in the context of crowdsourcing. We assume that two types, say, Type 1 and Type 2, of customers arrive to a c-server queueing system. A Type 1 customer has to receive service by one of c servers while a Type 2 customer may be served by a Type 1 customer who is available to act as a server soon after getting a service or by one of c servers. We assume that a Type 1 customer will be available for serving a Type 2 customer (provided there is at least one Type 2 customer waiting in the queue at the time of the service completion of that Type 1 customer) with probability \(p, 0 \le p \le 1\). With probability \(q = 1 - p\), a Type 1 customer will opt out of serving a Type 2 customer provided there is at least one Type 2 customer waiting in the system. Upon completion of a service a free server will offer service to a Type 1 customer on an FCFS basis; however, if there are no Type 1 customers waiting in the system, the server will serve a Type 2 customer if there is one present in the queue. If a Type 1 customer decides to serve a Type 2 customer, for our analysis purposes that Type 2 customer will be removed from the system as Type 1 customer will leave the system with that Type 2 customer. Under the assumption of exponential services for both types of customers we study the model in steady state using matrix analytic methods and establish some results including explicit ones for the waiting time distributions. Some illustrative numerical examples are presented.     

<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 type queueing-inventory systems with postponed work,reservation, cancellation and common life time

In this paper we analyze two single server queueing-inventory systems in which items in the inventory have a random common life time. On realization of common life time, all customers in the system are flushed out. Subsequently the inventory reaches its maximum level S through a (positive lead time) replenishment for the next cycle which follows an exponential distribution. Through cancellation of purchases, inventory gets added until their expiry time; where cancellation time follows exponential distribution. Customers arrive according to a Poisson process and service time is exponentially distributed. On arrival if a customer finds the server busy, then he joins a buffer of varying size. If there is no inventory, the arriving customer first try to queue up in a finite waiting room of capacity K. Finding that at full, he joins a pool of infinite capacity with probability γ (0 < γ < 1); else it is lost to the system forever. We discuss two models based on ‘transfer’ of customers from the pool to the waiting room / buffer. In Model 1 when, at a service completion epoch the waiting room size drops to preassigned number L ? 1 (1 < L < K) or below, a customer is transferred from pool to waiting room with probability p (0 < p < 1) and positioned as the last among the waiting customers. If at a departure epoch the waiting room turns out to be empty and there is at least one customer in the pool, then the one ahead of all waiting in the pool gets transferred to the waiting room with probability one. We introduce a totally different transfer mechanism in Model 2: when at a service completion epoch, the server turns idle with at least one item in the inventory, the pooled customer is immediately taken for service. At the time of a cancellation if the server is idle with none, one or more customers in the waiting room, then the head of the pooled customer go to the buffer directly for service. Also we assume that no customer joins the system when there is no item in the inventory. Several system performance measures are obtained. A cost function is discussed for each model and some numerical illustrations are presented. Finally a comparison of the two models are made.     

Analysis of <Emphasis Type="Italic">N</Emphasis>-policy queues with disastrous breakdown

In the area of optimal design and control of queues, the N-policy has received great attention. A single server queueing system with system disaster is considered where the server waits till N customers accumulate in the queue and upon the arrival of Nth customer the server begins to serve the customers until the system becomes idle or the occurrence of disaster whichever happens earlier. The system size probabilities in transient state are obtained in closed form using generating functions and steady-state system size probabilities are derived in closed form using generating functions and continued fractions. Further, the mean and variance for the number of customers in the system are derived for both transient and steady states and these results are deduced for the specific models. Time-dependent busy period distribution is also obtained. Numerical illustrations are also shown to visualize the system effect.     

Tables of Waiting Times for <Emphasis Type="Italic">M/M/n</Emphasis>, <Emphasis Type="Italic">M/D/n</Emphasis> and <Emphasis Type="Italic">D/M/n</Emphasis> and their Use to Give Approximate Waiting Times in More General Queues

An approximation formula for average waiting time in multiserver queues is considered using tables for the queues M/M/n, M/D/n and D/M/n. The approximation is compared with that proposed by Sakasegawa. Both approximations predominantly overestimate the waiting time, the first being more accurate, but the Sakasegawa approximation is simpler to apply.     

Heavy traffic limit for a tandem queue with identical service times

We consider a two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. We investigate the amount of work in the second queue at certain embedded arrival time points, namely when the upstream queue has just emptied. We focus on the case of infinite-variance service times and obtain a heavy traffic process limit for the embedded Markov chain.     

The Generalized <Emphasis Type="Italic">M/G/k</Emphasis> Blocking System with Heterogeneous Servers

This paper studies the equilibrium behaviour of the generalized M/G/k blocking system with heterogeneous servers. Such a service system has k servers, each with a (possibly) different service time distribution, whose customers arrive in accordance with a Poisson process. They are served, unless all the servers are occupied. In this case they leave and they do not return later (i.e. they are ‘blocked’). Whenever there are n (n = 0, 1, 2,..., k) servers occupied, each arriving customer balks with probability 1 - f _{n +1}(f _{k +1} = 0) and each server works at a rate g _n . Among other things, a generalization of the Erlang B-formula is given and also it is shown that the equilibrium departure process from the system is Poisson.     

Stability Criteria for Multi-class Queueing Networks with Re-entrant Lines

In this paper we show how the stability criteria for the model proposed in MacPhee and Müller (Queueing Syst 52(3):215–229, 2006) can be applied to queueing networks with re-entrant lines. The model considered has Poisson arrival streams, servers that can be configured in various ways, exponential service times and Markov feedback of completed jobs. The stability criteria are expressed in terms of the mean drifts of the process under the various server configurations. For models with re-entrant lines we impose here a boundary sojourn condition to ensure adequate control of the process when one or more queues are empty. We show with some examples, including the generalised Lu–Kumar network discussed in Niño-Mora and Glazebrook (J Appl Probab 37(3):890–899, 2000), how our results can be applied.     

Pull-based load distribution among heterogeneous parallel servers: the case of multiple routers

The model is a service system, consisting of several large server pools. A server’s processing speed and buffer size (which may be finite or infinite) depend on the pool. The input flow of customers is split equally among a fixed number of routers, which must assign customers to the servers immediately upon arrival. We consider an asymptotic regime in which the total customer arrival rate and pool sizes scale to infinity simultaneously, in proportion to a scaling parameter n, while the number of routers remains fixed. We define and study a multi-router generalization of the pull-based customer assignment (routing) algorithm PULL, introduced in Stolyar (Queueing Syst 80(4): 341–361, 2015) for the single-router model. Under the PULL algorithm, when a server becomes idle it sends a “pull-message” to a randomly uniformly selected router; each router operates independently—it assigns an arriving customer to a server according to a randomly uniformly chosen available (at this router) pull-message, if there is any, or to a randomly uniformly selected server in the entire system otherwise. Under Markov assumptions (Poisson arrival process and independent exponentially distributed service requirements), and under subcritical system load, we prove asymptotic optimality of PULL: as \(n\rightarrow \infty \), the steady-state probability of an arriving customer experiencing blocking or waiting vanishes. Furthermore, PULL has an extremely low router–server message exchange rate of one message per customer. These results generalize some of the single-router results in Stolyar (2015).     

Queueing with Cyclic Arrival Distributions and Variable Service Facilities: A Simulation Approach

This paper develops a near optimal policy for opening a second counter in a quick-service restaurant when n persons are in the first line, given that the arrival process is cyclic rather than purely stationary. A simulation was written in GPSS which permitted the use of different arrival generators during different time segments of the simulated day. Results indicate that as n is varied, waiting time and utilization levels exhibit an S-shaped curve between the pure 1 facility and the 2 facility cases. Results also indicate that a cyclic arrival process greatly increases (relative to the case of exponential arrivals) mean waiting time, facility utilization, and the range of n. Finding the optimal (least cost) value of n is illustrated for cases in which costs of customer waiting time can be meaningfully imputed.     

Large deviations for the total queue size in non-Markovian tandem queues

We consider a d-node tandem queue with arrival process and light-tailed service processes at all queues i.i.d. and independent of each other. We consider three variations of the probability that the number of customers in the system reaches some high level N, namely during a busy cycle, in steady state, and upon arrival of a new customer. We show that their decay rates for large N have the same value and give an expression for this value.     

<Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis> Queue with catastrophes and state-dependent control at idle time

We consider an M ^X /M/c queue with catastrophes and state-dependent control at idle time. Properties of the queues which terminate when the servers become idle are first studied. Recurrence, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no catastrophes. All of these properties and the first effective catastrophe occurrence time are then investigated for the case of resurrection and catastrophes. In particular, we obtain the Laplace transform of the transition probability for the absorbing M ^X /M/c queue.     

A Note on the Effectiveness of a Pool

A measure of ineffectiveness I of a pool of independent service facilities is defined and it isshown to be rather insensitive to changes in the service-time distribution for queues with aPoisson input, assuming equal mean service rates of the servers. This result reduces considerationto the simple case of exponentially distributed service times and in this case (i) I can be computedusing well-known formulae and (ii) it is possible to construct a simple upper bound for I.     

Multiset combinatorial batch codes

Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai, mimic a distributed storage of a set of n data items on m servers, in such a way that any batch of k data items can be retrieved by reading at most some t symbols from each server. Combinatorial batch codes, are replication-based batch codes in which each server stores a subset of the data items. In this paper, we propose a generalization of combinatorial batch codes, called multiset combinatorial batch codes (MCBC), in which n data items are stored in m servers, such that any multiset request of k items, where any item is requested at most r times, can be retrieved by reading at most t items from each server. The setup of this new family of codes is motivated by recent work on codes which enable high availability and parallel reads in distributed storage systems. The main problem under this paradigm is to minimize the number of items stored in the servers, given the values of n, m, k, r, t, which is denoted by N(n, k, m, t; r). We first give a necessary and sufficient condition for the existence of MCBCs. Then, we present several bounds on N(n, k, m, t; r) and constructions of MCBCs. In particular, we determine the value of N(n, k, m, 1; r) for any \(n\ge \left\lfloor \frac{k-1}{r}\right\rfloor {m\atopwithdelims ()k-1}-(m-k+1)A(m,4,k-2)\), where \(A(m,4,k-2)\) is the maximum size of a binary constant weight code of length m, distance four and weight \(k-2\). We also determine the exact value of N(n, k, m, 1; r) when \(r\in \{k,k-1\}\) or \(k=m\).     

Finite-buffer polling systems with threshold-based switching policy

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.     

On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues

This paper considers a particular renewal-reward process with multivariate discounted rewards (inputs) where the arrival epochs are adjusted by adding some random delays. Then, this accumulated reward can be regarded as multivariate discounted Incurred But Not Reported claims in actuarial science and some important quantities studied in queueing theory such as the number of customers in \(G/G/\infty \) queues with correlated batch arrivals. We study the long-term behaviour of this process as well as its moments. Asymptotic expressions and bounds for quantities of interest, and also convergence for the distribution of this process after renormalization, are studied, when interarrival times and time delays are light tailed. Next, assuming exponentially distributed delays, we derive some explicit and numerically feasible expressions for the limiting joint moments. In such a case, for an infinite server queue with a renewal arrival process, we obtain limiting results on the expectation of the workload, and the covariance of queue size and workload. Finally, some queueing theoretic applications are provided.     

Approximating the Mean Waiting Time in the <Emphasis Type="Italic">GI/G/s</Emphasis> Queue

We provide two distribution-dependent approximations for the mean waiting time in a GI/G/s queue. Both approximations are weighted combinations of the exact mean waiting times for the GI/M/s and M/D/s queues each of which has the same mean service time and traffic intensity as in the approximating GI/G/s queue. The weights in the approximations are expressed by the service-time c.d.f. and the first two moments of interarrival and service times. To examine the performance of our approximations, they are numerically compared with exact solutions and previous two-moment approximations for various cases. Extensive numerical comparisons indicate that the relative percentage errors of the approximations are of the order of 5% in moderate traffic and 1% in heavy traffic, except for extreme cases.