Random Number Of Servers Research Articles

Multiserver Queueing System with Constant Service Time and Simultaneous Service of a Customer by Random Number of Servers

This paper is focused on stability conditions of a multiserver queueing system with regenerative input flow where a random number of servers is simultaneously required for each customer, and each s...

Multiserver queueing system with constant service time and simultaneous service of a customer by random number of servers

Изучаются условия стабильности многоканальной системы обслуживания с регенерирующим входящим потоком, в которой одному требованию необходимо для обслуживания случайное число приборов одновременно и требование занимает каждый прибор на постоянное время. Оказывается, что условие стабильности зависит от интенсивности входящего потока, но не зависит от его структуры.

On Queues with a Random Capacity: Some Theory, and An Application (Technical Appendix)

Abstract from the main paper: One standard assumption in workforce management is that the firm can dictate to workers when to show up to work. However, that assumption is challenged in modern business environments, such as those arising in the sharing economy, where workers enjoy various degrees of flexibility, including the right to decide when to work. For example, a ride-sharing service cannot impose on its drivers to be on the road at specific times; similarly, a virtual call-center manager cannot direct her agents to be available for select shifts. When self-scheduling is allowed, the number of workers available in any time period is uncertain. In this chapter, we are concerned with the effective management of service systems where capacity, i.e., the number of available agents, is random. We rely on a queueing-theoretic framework, because customers are time-sensitive and delays are ubiquitous in the services industry, and focus on the performance analysis and control of a queueing system with a random number of servers. In particular, we begin by surveying some theoretical results on the control of queueing systems with uncertainty in parameters (here, the number of servers). Then, we illustrate how to apply those theoretical results to study the problems of staffing and controlling queueing systems with self-scheduling servers and impatient, time-sensitive, customers.

Flexible Workers or Full-Time Employees? On Staffing Systems with a Blended Workforce

There are different workforce models in the gig economy. While some on-demand service providers rely strictly on either traditional employees or independent contractors, others rely on a blended workforce which melds a layer of contingent workers with a core of permanent employees. In deciding on the number of right people to staff at the right time, managers must appropriately weigh the pertinent tradeoffs. In this paper, we study cost-minimizing staffing decisions in service systems where the manager must decide on how many flexible (contractors) and/or fixed (full-time) agents to staff in order to effectively balance operating costs, varying customer demand patterns, and supply-side uncertainty, while not compromising on the quality of service offered to customers. We consider a queueing-theoretic framework where the number of servers is random because part of the workforce is flexible. Since the staffing problem with a random number of servers is analytically intractable, we formulate two problem relaxations, based on fluid and stochastic-fluid formulations, and establish their accuracies in large systems by relying on an asymptotic, many-server, mode of analysis. We derive the optimal staffing policy and glean insights into the appropriateness of alternative workforce models in on-demand services. We also shed light on the distinction between demand-side (customer arrival rates) and supply-side (number of servers) uncertainties in queueing systems.

Capacity Sizing in Queueing Models with a Random Number of Servers

One prevalent assumption in queueing theory is that the number of servers in a queueing model is deterministic. However, randomness in the number of servers often arises in practice, particularly when the servers themselves may be viewed as strategic decision makers, e.g., as in virtual call centers or ride-sharing services where agents are allowed to set their own schedules, often at very short time notice. In this paper, we study the problem of staffing many-server queues with general abandonment and a random number of servers. We model the number of agents available to work in a given period as a binomial random variable, and rely on a fluid model to determine cost-minimizing staffing levels. We provide theoretical support for the usage of the fluid model by demonstrating the asymptotic accuracy of fluid-based performance measures and staffing prescriptions. As an application, we characterize the optimal staffing policy with self-scheduling servers, and show that it is not straightforward: It may be optimal to either understaff or overstaff the system, depending on (i) self-scheduling agent behavior and (ii) the abandonment-time distribution.

On the study of simultaneous service by random number of servers with retrial and preemptive priority

In this paper, we propose to study the performance of a multi-server queuing system in which a customer requires simultaneous service from a random number of servers with the queuing disciplines retrial and preemptive priority. The infinitesimal generator matrix is presented for the proposed model and steady state measures are discussed. In particular, analytical expressions are obtained for the case where a customer (single type) requires simultaneous service from a random number of servers. Also, the generalised stochastic Petri net (GSPN) is developed for the proposed model and the particular cases. The performance of the models is analysed and compared, in terms of average system size and throughput. It is found that queuing systems in which a customer requires simultaneous service from a random number of servers performs better when the queuing discipline of retrial is appended into the system.

A queueing system for modeling a file sharing principle

We investigate in this paper the performance of a simple file sharing principle. For this purpose, we consider a system composed of N peers becoming active at exponential random times; the system is initiated with only one server offering the desired file and the other peers after becoming active try to download it. Once the file has been downloaded by a peer, this one immediately becomes a server. To investigate the transient behavior of this file sharing system, we study the instant when the system shifts from a congested state where all servers available are saturated by incoming demands to a state where a growing number of servers are idle. In spite of its apparent simplicity, this queueing model (with a random number of servers) turns out to be quite difficult to analyze. A formulation in terms of an urn and ball model is proposed and corresponding scaling results are derived. These asymptotic results are then compared against simulations.

Bandwidth allocation in wide-band communication networks under two types of priorities

A wide-band ISDN provides access to many different users in a shared fashion. Control of the available bandwidth allocation is necessary for efficiency and flexibility. In this paper, the system is modelled as a multiple-server multiplequeue system, where access requests require service from a random number of servers. Requests are generated by a finite number of sources. Each request is entered in a queue so that the queue index indicates the bandwidth needed by the request (in basic bandwidth units). Control of service allocation is accomplished by assigning priorities to queues. Two priority types are examined here, where relatively higher priority is assigned to requests in the smaller indexed or the larger indexed queues. When more than one request can be served due to the availability of band-width, the one with the highest priority is chosen first. The procedures that regulate the bandwidth allocation are described, and a Markov-chain based solution to the model is developed to produce the system state probabilities, and utilization and performance measures.

Bandwidth Allocation Strategies in Wide-Band Integrated Networks

A common digital transmission facility in a wide-band integrated service digital network (ISDN) provides shared access to a community of heterogeneous users. Traffic demands from these users vary in their arrival rate, their service time, and their bit rate. In order for this type of communication system to handle its traffic demands with high efficiency and flexibility, a close control of access to the shared bandwidth is required. We model the system by a general multiserver queueing system where customers demand service from a random number of servers. If no waiting is allowed, this queueing model is readily analyzed, and various server allocation strategies can be studied. If the various access requests are queued for service, then the system calls for efficient strategies for allocating servers to waiting customers. In this case, exact analysis of the underlying queueing model becomes quite difficult. For this case, we present some analytic and simulation results of the performance of the system under several server allocation policies.

Technical Note—On Waiting Times for a Queue in Which Customers Require Simultaneous Service from a Random Number of Servers

We consider a queueing system, first introduced by L. Green in 1980, in which customers from a Poisson arrival stream request simultaneous service from a random number of identical servers with exponential service times. Computational formulas for the second moment of time in queue are given, along with tables of these values for selected systems. Numerical results show that the coefficient of variation for time in queue is always greater than 1 and decreases with increasing congestion.

Queues in Which Customers Receive Simultaneous Service from a Random Number of Servers: A System Point Approach

We examine a multi-server queueing system with Poisson arrivals in which customers require simultaneous service from a random number of servers. Servers assigned to the same customer begin and end service concurrently. Service times are, in general, assumed to be exponentially distributed. A system point approach is presented as a framework for obtaining the waiting time distribution for each customer type. Explicit solutions are derived for the two-server system.