Queues with paired customers

1981 ◽  
Vol 18 (3) ◽  
pp. 684-696 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Queueing Systems ◽  
Arrival Process ◽  
Service Rate ◽  
Probability Vector ◽  
Simple Manner ◽  
Special Service ◽  
Poisson Arrivals ◽  
The Difference ◽  
Number Of Customers ◽  
Stationary Probability Vector

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.

Queues with paired customers

1981 ◽  
Vol 18 (03) ◽  
pp. 684-696 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Queueing Systems ◽  
Arrival Process ◽  
Service Rate ◽  
Probability Vector ◽  
Simple Manner ◽  
Special Service ◽  
Poisson Arrivals ◽  
The Difference ◽  
Number Of Customers ◽  
Stationary Probability Vector

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.

On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions

1983 ◽  
Vol 20 (04) ◽  
pp. 860-871 ◽  
Author(s):  
Dieter König ◽  
Masakiyo Miyazawa ◽  
Volker Schmidt
Keyword(s):  
Stationary State ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Arrival Process ◽  
Loss Probabilities ◽  
Poisson Arrivals ◽  
Number Of Customers

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.

Some Pitfalls of Black Box Queue Inference: The Case of State-Dependent Server Queues

1993 ◽  
Vol 7 (2) ◽  
pp. 149-157 ◽  
Author(s):  
Sheldon M. Ross ◽  
J. George Shanthikumar ◽  
Xiang Zhang
Keyword(s):  
Queueing Systems ◽  
Work Load ◽  
Black Box ◽  
Arrival Process ◽  
Service Rate ◽  
Actual System ◽  
Sample Distribution ◽  
State Dependent ◽  
Service Times ◽  
Number Of Customers

In several queueing systems the service rate of a server is affected by the work load present in the system. For example, a teller at a bank or a checker at a check-out counter in a supermarket may change the service rate depending on the number of customers present in the system. But the service rate as a function of the number in the system can rarely be measured. Consequently, in a typical model of such a system it is assumed that the service rate is constant. Hence, such systems with a single stage are often modeled by GI/GI/c queueing systems with mutually independent arrival and service processes. Then the observed service times are used to find a sample distribution that will represent the distribution of the assumed i.i.d. service times. The purpose of this paper is to explore the effect of this black box queue inference (BBQI) in its ability to predict the performance of the actual system. In this regard, we have shown that when the arrival process is Poisson, if the servers react favorably [unfavorably] to higher work loads (i.e., if the server increases [decreases] the service rate as the number of customers in the system increases) then the BBQI predictions will be pessimistic [optimistic]. This result can be used to identify the server's attitude toward higher work load.

On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions

1983 ◽  
Vol 20 (4) ◽  
pp. 860-871 ◽  
Author(s):  
Dieter König ◽  
Masakiyo Miyazawa ◽  
Volker Schmidt
Keyword(s):  
Stationary State ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Arrival Process ◽  
Loss Probabilities ◽  
Poisson Arrivals ◽  
Number Of Customers

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

scholarly journals A discrete single server queue with Markovian arrivals and phase type group services

1995 ◽  
Vol 8 (2) ◽  
pp. 151-176 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
K. Laurie Dolhun ◽  
S. Chakravarthy
Keyword(s):  
Steady State ◽  
Queueing System ◽  
State Probability ◽  
Arrival Process ◽  
Single Server ◽  
Probability Vector ◽  
Phase Type Distribution ◽  
Steady State Probability ◽  
Phase Type ◽  
Number Of Customers

We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (1) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V(i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q(i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (01) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V (i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q (i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

Queueing output processes

1976 ◽  
Vol 8 (2) ◽  
pp. 395-415 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Queueing Systems ◽  
Second Order ◽  
Arrival Process ◽  
Single Server ◽  
Stationary Condition ◽  
Waiting Room ◽  
Stationary Point Process ◽  
Poisson Arrivals ◽  
Service Times ◽  
Markovian Systems

The paper reviews various aspects, mostly mathematical, concerning the output or departure process of a general queueing system G/G/s/N with general arrival process, mutually independent service times, s servers (1 ≦ s ≦ ∞), and waiting room of size N (0 ≦ N ≦ ∞), subject to the assumption of being in a stable stationary condition. Known explicit results for the distribution of the stationary inter-departure intervals {Dn} for both infinite and finite-server systems are given, with some discussion on the use of reversibility in Markovian systems. Some detailed results for certain modified single-server M/G/1 systems are also available. Most of the known second-order properties of {Dn} depend on knowing that the system has either Poisson arrivals or exponential service times. The related stationary point process for which {Dn} is the stationary sequence of the corresponding Palm–Khinchin distribution is introduced and some of its second-order properties described. The final topic discussed concerns identifiability, and questions of characterizations of queueing systems in terms of the output process being a renewal process, or uncorrelated, or infinitely divisible.

Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System

Processes ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 2146
Author(s):  
V. Vinitha ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
K. Jeganathan ◽  
Gyanendra Prasad Joshi ◽  
...  
Keyword(s):  
Steady State ◽  
Lead Time ◽  
Arrival Process ◽  
Inventory Level ◽  
Stationary Probability ◽  
Probability Vector ◽  
Steady State Analysis ◽  
Proposed Model ◽  
Stationary Probability Vector ◽  
Bernoulli Schedule

This article discusses the queueing-inventory model with a cancellation policy and two classes of customers. The two classes of customers are named ordinary and impulse customers. A customer who does not plan to buy the product when entering the system is called an impulse customer. Suppose the customer enters into the system to buy the product with a plan is called ordinary customer. The system consists of a pool of finite waiting areas of size N and maximum S items in the inventory. The ordinary customer can move to the pooled place if they find that the inventory is empty under the Bernoulli schedule. In such a situation, impulse customers are not allowed to enter into the pooled place. Additionally, the pooled customers buy the product whenever they find positive inventory. If the inventory level falls to s, the replenishment of Q items is to be replaced immediately under the (s, Q) ordering principle. Both arrival streams occur according to the independent Markovian arrival process (MAP), and lead time follows an exponential distribution. In addition, the system allows the cancellation of the purchased item only when there exist fewer than S items in the inventory. Here, the time between two successive cancellations of the purchased item is assumed to be exponentially distributed. The Gaver algorithm is used to obtain the stationary probability vector of the system in the steady-state. Further, the necessary numerical interpretations are investigated to enhance the proposed model.

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